Unintegrated gluon distribution from modified BK equation

TL;DR

The paper develops and numerically analyzes a modified BK equation for the unintegrated gluon distribution that incorporates subleading small-x corrections via a unified BFKL-DGLAP kernel, a kinematic constraint, running coupling, and a nonlinear saturation term. It shows that subleading corrections notably reduce the intercept and overall normalization, with nonlinear effects becoming important even in the nominally dilute regime, leading to significant differences in the integrated gluon density xg(x,Q^2). The saturation scale Qs(x) derived from the model is consistent with GBW estimates, while a relative-difference measure suggests saturation effects can emerge at higher x than the conventional Qs would indicate. The dipole cross section σ(r,x) extracted from the model closely matches the GBW prediction, supporting the approach, though a complete NLLx treatment would require including nonlinear effects in the triple-Pomeron vertex.

Abstract

We investigate the recently proposed nonlinear equation for the unintegrated gluon distribution function which includes the subleading effects at small $x$. We obtained numerically the solution to this equation in $(x,k)$ space, and also the integrated gluon density. The subleading effects affect strongly the normalization and the $x$ and $k$ dependence of the gluon distribution. We show that the saturation scale $Q_s(x)$, which is obtained from this model, is consistent with the one used in the saturation model by Golec-Biernat and Wüsthoff. We also estimate the nonlinear effects by looking at the relative normalization of the solutions to the linear and nonlinear equations. It turns out that the differences are quite large even in the nominally dilute regime, that is when $Q^2 \gg Q_s^2$. Finally, we calculate the dipole-nucleon cross section.

Figures (7)

• Figure 1: The unintegrated gluon distribution $f(x,k^2)$ obtained from Eq.(\ref{['eq:kovev']}) as a function of $x$ for different values $k^2 = 10^2 \ \hbox{\rm GeV}^2$ and $k^2 = 10^3 \ \hbox{\rm GeV}^2$ (left) and for $k^2 = 5 \ \hbox{\rm GeV}^2$ and $k^2 = 50 \ \hbox{\rm GeV}^2$ (right) . The solid lines correspond to the solution of the nonlinear equation (\ref{['eq:kovev']}) whereas the dashed lines correspond to the linear BFKL/DGLAP term in (\ref{['eq:kovev']}).
• Figure 2: The same as Fig. \ref{['fig:fx']} but now the modifed BK equation (\ref{['eq:kovev']}) (solid lines) is compared with the original BK equation (\ref{['eq:modf']}) without subleading corrections (dashed lines).
• Figure 3: The unintegrated gluon distribution $f(x,k^2)$ as a function of $k^2$ for two values of $x=10^{-5}$ and $10^{-4}$ . Left: solid lines correspond to the solution of the nonlinear equation (\ref{['eq:kovev']}) whereas dashed lines correspond to linear BFKL/DGLAP term in (\ref{['eq:kovev']}). Right: solid lines correspond to the solution of the nonlinear equation (\ref{['eq:kovev']}) whereas dashed lines correspond to the solution of the original BK equation without the NLLx modifications in the linear part (\ref{['eq:modf']}).
• Figure 4: The integrated gluon distribution $xg(x,k^2)$ as a function of $x$ for values of $Q^2 = 10^2 \hbox{\rm GeV}^2$ and $Q^2 = 10^3$ (left) and for $Q^2 = 5 \ \hbox{\rm GeV}^2$ and $Q^2 = 50$ (right) obtained from integrating $f(x,k^2)$Eq.(\ref{['eq:kovev']}). Dashed lines correspond to solution of linear BFKL/DGLAP evolution equation.
• Figure 5: Saturation scale obtained from Eqs. (\ref{['eq:qsats']},\ref{['eq:omegas']}) solid lines, compared with saturation scale from GBW model GBW.