Solving the high energy evolution equation including running coupling corrections

We study the solution of the nonlinear Balitsky-Kovchegov evolution equation with the recently calculated running coupling corrections [I. I. Balitsky, Phys. Rev. D 75, 014001 (2007). and Y. Kovchegov and H. Weigert, Nucl. Phys. A784, 188 (2007).]. Performing a numerical solution we confirm the earlier result of Albacete et al. [Phys. Rev. D 71, 014003 (2005).] (obtained by exploring several possible scales for the running coupling) that the high energy evolution with the running coupling leads to a universal scaling behavior for the dipole-nucleus scattering amplitude, which is independent of the initial conditions. It is important to stress that the running coupling corrections calculated recently significantly change the shape of the scaling function as compared to the fixed coupling case, in particular, leading to a considerable increase in the anomalous dimension and to a slow-down of the evolution with rapidity. We then concentrate on elucidating the differences between the two recent calculations of the running coupling corrections. We explain that the difference is due to an extra contribution to the evolution kernel, referred to as the subtraction term, which arises when running coupling corrections are included. These subtraction terms were neglected in both recent calculations. We evaluate numerically the subtraction terms for both calculations, and demonstrate that when the subtraction terms are added back to the evolution kernels obtained in the two works the resulting dipole amplitudes agree with each other. We then use the complete running coupling kernel including the subtraction term to find the numerical solution of the resulting full nonlinear evolution equation with the running coupling corrections. Again the scaling regime is recovered at very large rapidity with the scaling function unaltered by the subtraction term