The Gluon Green's Function in the BFKL Approach at Next-to-Leading Logarithmic Accuracy

TL;DR

The paper develops and tests an iterative numerical solution to the BFKL equation at next-to-leading logarithmic (NLL) accuracy, preserving full angular dependence and running coupling effects by solving the equation as originally formulated. Using a Monte Carlo implementation with phase-space slicing, it demonstrates convergence, infrared finiteness, and detailed behavior of the gluon Green's function as a function of external momenta, energy evolution (Δ), and angular correlations, complemented by a toy cross section. The results show that NLL corrections reduce the intercept and reduce renormalisation-scale uncertainty, indicating improved predictive power and stability relative to LL, while highlighting the influence of scheme choices for the running coupling. Overall, the method proves feasible and informative for exploring high-energy QCD dynamics at NLL, with clear avenues for further refinements in running-coupling schemes.

Abstract

We investigate the gluon Green's function in the high energy limit of QCD using a recently proposed iterative solution of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation at next-to-leading logarithmic (NLL) accuracy. To establish the applicability of this method in the NLL approximation we solve the BFKL equation as originally written by Fadin and Lipatov, and compare the results with previous studies in the leading logarithmic (LL) approximation.

Figures (10)

• Figure 1: Comparison of the gluon Regge trajectory calculated at LL and at NLL.
• Figure 2: Structure of the kernel $\tilde{K}_r \left(q,q',\theta\right)$ (in GeV$^{-2}$) for $q' = 20$ GeV as a function of $q$ and the angle between ${\bf q}$ and ${\bf q'}$.
• Figure 3: Distribution on the number of iterations in building up the NLL gluon Green's function for different values of the parameter $\Delta$, at a fixed value of $\lambda=1$ GeV. The gluon Green's function is evaluated for $k_a=25$ GeV, $k_b=30$ GeV and the renormalisation scale is chosen to be $\mu=k_b$.
• Figure 4: Dependence of the NLL solution for the gluon Green's function on the parameter $\lambda$ for $k_a=25$ GeV, $k_b=30$ GeV and $\Delta=3$.
• Figure 5: $k_a$ dependence of the LL and NLL gluon Green's function at $\mu=k_b=30$ GeV for two values of $\Delta$.