Unintegrated gluon distributions from the transverse coordinate representation of the CCFM equation in the single loop approximation

TL;DR

The paper analyzes unintegrated gluon distributions within the CCFM framework in the single-loop approximation. By diagonalizing the equation in the transverse-coordinate space with a Fourier-Bessel transform, it derives an analytic solution for the moment function and expresses unintegrated distributions in terms of the integrated gluon distribution through tractable approximations. The work also provides a numerical study comparing the exact $b$-space solution to two approximate formulations, highlighting where these approximations hold and where they fail. This approach offers analytic insight into the CCFM dynamics, connects to LO DGLAP behavior at small scales, and suggests a path toward extending the method beyond the single-loop approximation.

Abstract

We utilise the fact that the CCFM equation in the single loop approximation can be diagonalised by the Fourier-Bessel transform. The analytic solution of the CCFM equation for the moments $f_ω(b,Q)$ of the scale dependent gluon distribution is obtained, where $b$ is the transverse coordinate conjugate to the transverse momentum of the gluon. The unintegrated gluon distributions obtained from this solution are analysed. It is shown how the approximate treatment of the exact solution makes it possible to express the unintegrated gluon distributions in terms of the integrated ones. The corresponding approximate expressions for the unintegrated gluon distribution are compared with exact solution of the CCFM equation in the single loop approximation.

Figures (3)

• Figure 1: Function $Q_t^2f(x,Q_t,Q)$, where $f(x,Q_t,Q)$ is the unintegrated gluon distribution obtained from the exact solution of the CCFM equation in the single loop approximation, plotted as the function of the transverse momentum $Q_t$ of the gluon for $Q^2=100 GeV^2$. The upper and lower curves correspond to $x=0.01$ and $x=0.1$ respectively. The transverse momentum $Q_t$ is in $GeV$.
• Figure 2: Function $Q_t^2f(x,Q_t,Q)$, where $f(x,Q_t,Q)$ is the unintegrated gluon distribution obtained from the exact solution of the CCFM equation in the single loop approximation, plotted as the function of the transverse momentum $Q_t$ of the gluon for $Q^2=100 GeV^2$ and $x=0.01$. The solid curve corresponds to $f(x,Q_t,Q)$ obtained from exact solution of the CCFM equation in the single loop approximation, while the short dashed and long dashed curves correspond to approximate expressions for $f(x,Q_t,Q)$ given by equations (34) and (36) respectively. The transverse momentum $Q_t$ is in $GeV$.
• Figure 3: Function $Q_t^2f(x,Q_t,Q)$, where $f(x,Q_t,Q)$ is the unintegrated gluon distribution obtained from the exact solution of the CCFM equation in the single loop approximation, plotted as the function of the transverse momentum $Q_t$ of the gluon for $Q^2=100 GeV^2$ and $x=0.1$. The solid curve corresponds to $f(x,Q_t,Q)$ obtained from the exact solution of the CCFM equation in the single loop approximation, while the short dashed and long dashed curves correspond to approximate expressions for $f(x,Q_t,Q)$ given by equations (34) and (36) respectively. The transverse momentum $Q_t$ is in $GeV$.