Transverse momentum dependent gluon density in a proton at low $x$ in the Laplace transform method

Abstract

We investigate the gluon distribution in a proton at very low $x$, both integrated and transverse momentum dependent, using the Laplace transform technique. By accounting for leading and main next-to-leading contributions, we derive compact analytical expressions for the gluon densities valid in the asymptotic limit $x \to 0$. Our results closely match those from other analytical and numerical approaches, with the main advantage being the simplicity of the expressions, which capture the essential features of more complex calculations.

Figures (3)

• Figure 1: Conventional gluon density in a proton $xg(x,\mu^2)$ calculated at the LO as function of $x$ for different values of $\mu^2$. For comparison we show here the results of numerical solutions of the LO DGLAP equations performed by the CTEQ-TEA CT14, NNPDF4.0 NNPDF4, MSHT'2020 MSHT20 and IMP IMP groups.
• Figure 2: Conventional gluon density in a proton $xg(x,\mu^2)$ calculated at the NLO as function of $x$ for different values of $\mu^2$. For comparison we show here the results of numerical solutions of the NLO DGLAP equations performed by the CTEQ-TEA CT18, NNPDF4.0 NNPDF4 and MSHT'2020 MSHT20 groups.
• Figure 3: TMD gluon densities in a proton $f_g(x, {\mathbf k}_T^2)$ calculated as function of ${\mathbf k}_T^2$ for different values of $x$. For comparison we show corresponding results obtained within the KMR/WMR and CCFM approaches, labeled as KL'2025 TMDs-appr5 and LLM'2024 TMDs-appr2, respectively. Note that we set $\mu^2 = 100$ GeV$^2$ for two-scale involved gluon distributions KL'2025, LLM'2024 and the one given by (\ref{['eq-improved-gluon']}).