this equation do not explicitly depend on the scale $\mu$ and that the new form of the initial condition has to be used (this is denoted by a tilde sign).

4.  BFKL with the kinematic constraint and DGLAP correction (BFKL+CD)

    In Ref. [@Kwiecinski:1997ee] yet another improvement of (\[eq:BFKL\]) was proposed. One can make an attempt to account for DGLAP-like behaviour by including the non-singular part of the gluon splitting function (the third term below) $$\begin{gathered}
    \mathcal{F}\left(x,k_{T}^{2}\right)=\mathcal{F}_{0}\left(x,k_{T}^{2}\right)\\
    +\overline{\alpha}_{s}\left(k_{T}^{2}\right)\int_{x}^{1}\frac{dz}{z}\,\int_{k_{T0}^{2}}^{\infty}dq_{T}^{2}\left[\frac{q_{T}^{2}\mathcal{F}\left(\frac{x}{z},q_{T}^{2}\right)\Theta\left(k_{T}^{2}-zq_{T}^{2}\right)-k_{T}^{2}\mathcal{F}\left(\frac{x}{z},q_{T}^{2}\right)}{\left|q_{T}^{2}-k_{T}^{2}\right|}+\frac{k_{T}^{2}\mathcal{F}\left(\frac{x}{z},k_{T}^{2}\right)}{\sqrt{4q_{T}^{4}+k_{T}^{4}}}\right]\\
    +\overline{\alpha}_{s}\left(k_{T}^{2}\right)\int_{x}^{1}\frac{dz}{z}\,\left(\frac{z}{2N_{c}}P_{gg}\left(z\right)-1\right)\int_{k_{T0}^{2}}^{k_{T}^{2}}dq_{T}^{2}\mathcal{F}\left(\frac{x}{z},q_{T}^{2}\right),\label{eq:BFKL+CD}\end{gathered}$$ where $P_{gg}\left(z\right)$ is the standard gluon splitting function. This correction, similar to the kinematic constraint, accounts for certain sub-leading corrections to the BFKL equation.

5.  BFKL with DGLAP correction alone\[enu:lastmodel\]

    This variant is used to test the significance of the DGLAP term alone.

The above UGDs do not involve any hard scale dependence. For observables involving high-$p_{T}$ jets a presence of large scale $\mu^{2}\sim p_{T}^{2}$ in perturbative calculations would involve additional logarithms of the type $\log\left(\mu^{2}/k_{T}^{2}\right)$ which can spoil the procedure. Therefore a re-summation of those logs is desired and it accounts in hard scale dependence for UGDs, c.f. Eq. (\[eq:HEN\_fact\_2\]). The approach which incorporates both $x$, $k_{T}^{2}$ and $\mu^{2}$ dependence in UGDs is provided for example by the CCFM evolution equation 