[Paper Review] Unintegrated gluon distributions from the transverse coordinate representation of the CCFM equation in the single loop approximation

J. Kwieciǹski|ArXiv.org|Mar 18, 2002

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper presents an exact analytical solution for the unintegrated gluon distribution using the transverse coordinate representation of the CCFM equation in the single-loop approximation. By applying the Fourier-Bessel transform to diagonalize the equation, it derives exact expressions for the moment function $ f_\omega(b,Q) $, which are then used to analyze and validate approximate relations between unintegrated and integrated gluon distributions, showing good agreement except near boundary regions of transverse momentum.

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.

Motivation & Objective

• To analytically solve the CCFM equation in the single-loop approximation using transverse coordinate representation.
• To derive unintegrated gluon distributions from the exact solution via Fourier-Bessel transform.
• To examine and validate approximate formulas linking unintegrated and integrated gluon distributions in the DGLAP regime.
• To compare the exact solution with approximate expressions numerically, particularly in the phenomenologically relevant region $ x \geq 0.01 $.

Proposed method

• The CCFM equation is transformed into transverse coordinate space using the Fourier-Bessel transform, which diagonalizes the equation in the single-loop approximation.
• The moment function $ f_\omega(b,Q) $ is solved analytically in transverse coordinate space, where $ b $ is conjugate to transverse momentum $ Q_t $.
• Unintegrated gluon distributions are reconstructed via inverse Fourier-Bessel transform of the moment function.
• The solution is compared with approximate expressions derived from DGLAP dynamics, particularly those relating $ f(x,Q_t,Q) $ to the integrated gluon distribution $ g(x,Q^2) $.
• Numerical results are generated using an input non-perturbative distribution $ \bar{f}^0(x,b) = \frac{g_0(x)}{2} \exp(-b^2 q_0^2 / 4) $ with $ g_0(x) = 3(1-x)^5 $ and $ q_0 = 1\,\text{GeV} $.
• The validity of approximate formulas is assessed by comparing them with the exact solution across different $ x $ and $ Q_t $ values.

Experimental results

Research questions

• RQ1 Can the CCFM equation in the single-loop approximation be exactly solved in transverse coordinate space using the Fourier-Bessel transform?
• RQ2 How do approximate formulas for unintegrated gluon distributions, based on DGLAP evolution, emerge from the exact solution?
• RQ3 What is the quantitative accuracy of these approximate formulas compared to the exact solution across different $ x $ and $ Q_t $?
• RQ4 How do the approximations perform near the boundaries $ Q_t^2 \sim Q_0^2 $ and $ Q_t^2 \sim Q^2 $, where exact solutions show deviations?

Key findings

• The CCFM equation in the single-loop approximation is exactly solvable in transverse coordinate space via the Fourier-Bessel transform, yielding an analytical expression for the moment function $ f_\omega(b,Q) $.
• The exact solution allows for the derivation of unintegrated gluon distributions through inverse Fourier-Bessel transform, providing a rigorous benchmark for approximate models.
• Equation (36), an improved approximate form, shows better agreement with the exact solution than the simpler formula (34), especially in the intermediate $ Q_t $ range.
• Formula (34) provides a reasonable approximation for small $ Q_t $, but can yield unphysical negative values at large $ x \sim 0.1 $ and large $ Q_t^2 $.
• The approximation (36) performs well except near the endpoints $ Q_t^2 \sim Q_0^2 $ and $ Q_t^2 \sim Q^2 $, where the exact solution exhibits non-trivial behavior.
• The study confirms that the transverse coordinate representation is a powerful tool for analytical insight into the CCFM equation in the single-loop regime, with potential for broader applicability beyond this approximation.