The evolution of the small x gluon TMD

TL;DR

This work addresses how to resum two distinct large logarithms in small-x gluon dynamics within the overlap region of TMD factorization and high-energy factorization. Using a simple quark target model in the Ji-Ma-Yuan scheme, it computes the small-x gluon TMD at NLO and demonstrates that its evolution satisfies both Collins-Soper and BFKL equations, supporting a unified description in the dilute limit. The main contributions are the explicit separation and resummation of ln(1/x) and ln( x^2 ζ^2 / l_⊥^2 ) logs via real and virtual corrections, establishing a concrete link between TMDs and unintegrated gluon distributions in the overlap region. This provides a framework for joint resummation of S/M^2 and M^2/p_⊥^2 logarithms in processes with well-separated scales, and points to future extensions to saturation effects and polarized observables.

Abstract

We study the evolution of the small $x$ gluon transverse momentum dependent(TMD) distribution in the dilute limit. The calculation has been carried out in the Ji-Ma-Yuan scheme using a simple quark target model. As expected, we find that the resulting small $x$ gluon TMD simultaneously satisfies both the Collins-Soper(CS) evolution equation and the Balitsky-Fadin-Kuraev-Lipatov(BFKL) evolution equation. We thus confirmed the earlier finding that the high energy factorization(HEF) and the TMD factorization should be jointly employed to resum the different type large logarithms in a process where three relevant scales are well separated.

Figures (4)

• Figure 1: The leading order diagram contributing to the gluon TMD in the quark target model. $P$ and $l$ are the incoming quark and the outgoing gluon momenta, respectively. $i$, $j$ denote the gluon polarization indices which will be contracted with $\delta_\perp^{ij}$ in the unpolarized case.
• Figure 2: Sample real diagrams contributing to the evolution kernels of the CS equation and the BFKL equation. All real diagrams shown here contribute to the evolution kernel of the BFKL equation. The Fig.(a) and its conjugate diagram are the only real corrections that contribute to both the evolution kernels of the BFKL equation and the CS equation.
• Figure 3: In the strong rapidity ordering region $l^+ \ll k^+ \ll P^+$, all real corrections Fig.(a)-Fig.(e) can be summarized into one diagram Fig.(f) with an effective Lipatov vertex. The large solid circle denotes the Lipatov vertex.
• Figure 4: Virtual diagrams contributing to the gluon TMD at NLO. The Fig.a and Fig.b and their conjugate diagrams give rise to the virtual corrections to the BFKL evolution kernel, while the Fig.c and its conjugate diagram contribute to the CS evolution kernel.