Triumvirate of Running Couplings in Small-x Evolution

TL;DR

This work addresses the absence of running-coupling corrections in the non-linear small-x evolution kernels JIMWLK and BK by performing a diagrammatic resummation of fermion bubbles within light-cone perturbation theory. Through UV subtraction and a scheme-dependent subtraction point, the authors extract the running-coupling contributions and demonstrate a consistent all-orders resummation that yields a running-coupling kernel in momentum space with a distinctive triumvirate structure of couplings. The coordinate-space formulation introduces an explicit scale R(x0,x1;z) that governs the three running couplings, and the results agree with the dispersive approach of Gardi (2006) while providing a transparent BLM-style interpretation. The findings offer a perturbatively controlled framework for small-x evolution, clarifying scheme dependence and enabling rc-JIMWLK and rc-BK equations applicable to high-density QCD processes at small x.

Abstract

We study the inclusion of running coupling corrections into the non-linear small-x JIMWLK and BK evolution equations by resumming all powers of alpha_s N_f in the evolution kernels. We demonstrate that the running coupling corrections are included in the JIMWLK/BK evolution kernel by replacing the fixed coupling constant alpha_s in it with alpha_s (1/r_1^2) alpha_s (1/r_2^2) / alpha_s (1/R^2), where r_1 and r_2 are transverse distances between the emitted gluon and the harder gluon (or quark) off of which it was emitted to the left and to the right of the interaction with the target. In the formalism of Mueller's dipole model r_1 and r_2 are the transverse sizes of ``daughter'' dipoles produced in one step of the dipole evolution. The scale R is a function of two-dimensional vectors r_1 and r_2, the exact form of which is scheme-dependent. We propose using a particular scheme which gives us R as an explicit function of r_1 and r_2.

Figures (6)

• Figure 1: Diagrams giving the leading $\alpha_s N_f$ correction to the kernels of JIMWLK and BK small-$x$ evolution equations. The thick dots on gluon and quark lines denote interactions with the target.
• Figure 2: Diagrams with instantaneous parts of gluon propagators giving the leading $\alpha_s N_f$ correction to the kernels of JIMWLK and BK small-$x$ evolution equations. There is no analog of Fig. \ref{['fig:NLO1']} B. All the lines are implied to be labeled in the same way as in Fig. \ref{['fig:NLO1']}.
• Figure 4: Separating UV-finite and UV-divergent parts of the kernel $K_1^{\text{NLO}}$ in the NLO BK evolution. The ovals denote color dipoles.
• Figure 5: Diagrams giving the higher order $\alpha_s N_f$ corrections to the kernels of JIMWLK and BK small-$x$ evolution equations. To get the all-order $\alpha_s N_f$ contribution one has to sum an infinite series of quark bubble insertions.
• Figure 6: Diagrams giving the higher order $\alpha_s N_f$ corrections to the kernels of JIMWLK and BK small-$x$ evolution equations containing instantaneous gluon lines. Again, to get the all-order $\alpha_s N_f$ contribution one has to sum an infinite series of quark bubble insertions.