In the case of gluons (i.e a process where a gluon splits into two gluons) the number of gluons is not conserved but the total energy is, therefore, in this case, the integral of $z\hat{P}$ over $z$ is zero.

### BFKL region

Assume that $Q^2$ is only moderately large and that $x$ is kept small. The splitting function for the process where a gluon splits into two gluons, $P_{gg}$, contains a pole, $1/z$. This implies that for small $x$ and moderately large $Q^2$, this process is very probable and will dominate over splitting events involving quarks. Therefore we now consider gluonic chains where a parton emits a gluon which then develops a chain of gluons by successive splittings into new gluons. In this region, non-ordered ladders, though suppressed, will be more important and they have to be taken into account. This means that the transverse momenta, $k_\perp$, can now also decrease. Considering this new possibility, Balistky, Fadin, Kuraev and Lipatov derived a more complex evolution equation, which is known as the BFKL equation. The evolution in $x$ is given by $$\frac{\partial \mathcal{F}(x,k_{\perp}^2)}{\partial \textrm{ln}(1/x)} = \int dk_{\perp}^{'2}
K(k_{\perp}^2,k_{\perp}^{'2})\mathcal{F}(x,k_\perp^{'2}) 
\label{eq:BFKL}$$ Here we have introduced the unitegrated distribution function, $\mathcal{F}$. There are various 