On the running coupling in the JIMWLK equation

TL;DR

This work introduces a novel running-coupling implementation for the JIMWLK equation by tying the coupling to the momentum scale of emitted gluons through the noise correlator in the Langevin formulation. The authors connect this rcJIMWLK approach to the established Balitsky rcBK prescription, derive BK-like behavior in appropriate limits, and provide detailed numerical comparisons that show a slower evolution than the simpler square-root rcJIMWLK but not identical to rcBK. The method leverages FFT-based efficiency to modify the noise correlator without altering the core JIMWLK kernel, and reveals how the parent and daughter dipole scales govern the running coupling. Overall, the proposal offers a practical and physically motivated way to implement running coupling in JIMWLK with implications for closer alignment with HERA data, while highlighting lattice-related uncertainties and the need for continuum studies.

Abstract

We propose a new method to implement the running coupling constant in the JIMWLK equation, imposing the scale dependence on the correlation function of the random noise in the Langevin formulation. We interpret this scale choice as the transverse momentum of the emitted gluon in one step of the evolution and show that it is related to the "Balitsky" prescription for the BK equation. This slows down the evolution speed of a practical solution of the JIMWLK equation, bringing it closer to the x-dependence inferred from fits to HERA data. We further study our proposal by a numerical comparison of the BK and JIMWLK equations.

Figures (11)

• Figure 1: Virtual diagrams.
• Figure 2: Real diagrams.
• Figure 3: Virtual diagrams in Fig. \ref{['fig:virt']} rewritten with the explicit momentum argument used as the scale of the running coupling.
• Figure 4: Real diagrams in Fig. \ref{['fig:real']} rewritten with the explicit momentum argument used as the scale of the running coupling.
• Figure 5: Evolution of the scattering amplitude with the JIMWLK equation where the scale of the running coupling is set in the noise term, Eq. (\ref{['eq:newcorr']}), compared to BK evolution with the "Balitsky" running coupling prescription, Eq. (\ref{['eq:bal']}). The evolution is started from the same initial condition and the lines show the amplitude at intervals of 2 units in rapidity.