Gluon distributions from Oliveira-Martin-Ryskin combined $\mathrm{BFKL}+\mathrm{DGLAP}$ evolution equations

Kwiecinski, Martin, and Stasto [Phys. Rev. D 56, 3991 (1997)] argue for inclusion of DGLAP terms into BFKL evolution of unintegrated gluon density. The equation was reformulated by Oliveira, Martin, and Ryskin [Evolution in Opening Angle Combining DGLAP and BFKL Logarithms (to be published)] employing the opening angle $\theta =\frac{k}{xp}$ as the evolution variable. It leads to a description of a $\theta$-integrated gluon density. This paper is a numerical study of these two similar combined $\mathrm{BFKL}+\mathrm{DGLAP}$ formulations. It is a demonstration of feasibility of the new approach. The different ways of subtracting the contribution common for BFKL and DGLAP proposed by Kwiecinski, Martin, and Stasto [Phys. Rev. D 56, 3991 (1997)] and Oliveira, Martin, and Ryskin [Evolution in Opening Angle Combining DGLAP and BFKL Logarithms (to be published)] are compared. The numerical tests confirm that the $\theta$ variable is a more natural evolution variable for this kind of equation.

Figure 1

Depiction of action of the kernel for a particular $(x,k)$ point on the $\log x,\log k$ plane (left) and $\log x,\log \theta$ plane (right). The highlighted lines have the following meaning: (a) infrared cutoff, (b) a line of constant $x$, which contributes to the virtual DGLAP terms, (c) a cut imposed by kinematical consistency requirement, (d) the $k_{\max }$ cutoff applied for numerical convenience, (e) (pink line) the points contributing to the virtual terms of the BFKL kernel, (f) a line of constant $\theta$. Arrows: (g) the usual evolution with $x$, (h) the new evolution directed by $\theta$; blue area: the points contributing to real emissions of BFKL.

Figure 2

Gluon distributions $\mathcal{F}(x,k^2)$ resulting from: Eq. (6) with the $I_{dc}^{\mathrm{DGLAP}}$ modification ($a$ and $c$); Eq. (1) ($b$ and $c$). Lines $c$ show a correctly converged result; $a$ and $b$ are obtained with the algorithm limited to 8 refinements to compare the behaviour of the solvers.