Small-x TMD distributions initial condition: Nc-dependence and Gaussian approximations

We systematically derive expressions for ten small-$x$ Transverse Momentum Dependent (TMD) distributions in the Gaussian approximation, three in the quark-gluon sector and seven in the gluon-gluon sector; for the general $SU(N_c)$ gauge group. The derived formulae depend on the logarithm of the dipole amplitude. Using the McLerran-Venugopalan model for the initial condition, we simulate the dipole amplitude as well as estimate all ten TMD distributions for $N_c \in \{2,3,4,5\}$. We compare these explicit numerical results with the derived expressions and find very good agreement for all studied values of $N_c$. Consequently, we study the scaling with $N_c$ of the TMD distributions. Thanks to that, we are able to derive the large-$N_c$ limit of the Gaussian approximation results and show that they agree with expressions obtained in the mean-field approximation. By comparing with our numerical results, we are able to demonstrate the size, origin, and significance of subleading-$N_c$ corrections. This work sets the stage for a systematic study of subleading corrections induced by the JIMWLK evolution in the rapidity evolution of the TMD distributions. Interestingly, we find an exact sum rule that relates all seven gluon-gluon TMD operators at $N_c = 3$ and arbitrary $x$.