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

A

bstract

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 ∈ {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 .