On the small-x evolution of the color quadrupole and the Weizs\"acker-Williams gluon distribution

Color quadrupoles have been found to be important in the proper description of observables sensitive to the small-x regime in nuclei as well as in the operator definition of the Weizs\"acker-Williams gluon distribution. In this paper, we derive the small-x evolution equation of the quadrupole and the Weizs\"acker-Williams gluon distribution without taking large N_c limit and study the properties of the equation in both dilute and saturation regime. We find that the quadrupole evolution follows the BFKL evolution in the dilute regime and then saturates in the dense region due to nonlinear terms. This leads us to conclude that the Weizs\"acker-Williams gluon distribution should obey the same geometrical behavior as the dipole gluon distribution as found in the inclusive DIS measurement.