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

TL;DR

This paper derives and numerically tests small-x TMD distributions within the Gaussian approximation for various Nc values, exploring Nc-dependence, large-Nc limit, and subleading corrections, with implications for rapidity evolution studies.

Contribution

It provides explicit formulas for small-x TMD distributions for general SU(Nc), compares numerical simulations with theoretical expressions, and analyzes Nc scaling and subleading corrections.

Findings

Good agreement between numerical and analytical results for all Nc values studied.

Derived large-Nc limit matches mean-field approximation results.

Identified an exact sum rule relating gluon-gluon TMD operators at Nc=3.

Abstract

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 $x$.