variable ln$1/x$, $F(x,Q^2)$ for fermions and $G(x,Q^2)$ for gluons.

DGLAP, BFKL and the LDC Models
------------------------------

### DGLAP evolution

Consider now the emission of many gluons before the parton is hit by the photon. This is represented by a fan diagram, see figure 2.

(200,210)(0,0) (0,0)(20,30) (20,30)(30,60) (30,60)(36,90) (36,90)(40,120) (40,120)(40,150) (40,150)(35,180) (0,210)(35,180)[4]{}[6]{} (20,30)(70,30)[4]{}[5]{} (30,60)(80,60)[4]{}[5]{} (36,90)(86,90)[4]{}[5]{} (40,120)(90,120)[4]{}[5]{} (40,150)(90,150)[4]{}[5]{} (35,180)(85,180) (5,20)\[\][$k_0$]{} (20,50)\[\][$k_1$]{} (28,80)\[\][$k_2$]{} (30,170)\[\][$k_n$]{} (85,30)\[\][$q_1$]{} (95,60)\[\][$q_2$]{} (105,150)\[\][$q_n$]{}

We define $z_i$ as, $z_i=k_{+i}/k_{+i-1}$, hence $1-z_i$ is the fraction of positive light cone momentum, $p_+=p_0+p_l$, carried away by the quasi-real gluon, $q_i$. This implies that the Bjorken variable, $x$, is given by $x=\prod_{i=1}^n
z_ix_0$. In the double leading log approximation the gluon ladder is strongly ordered in transverse momentum $$q_{\perp 1}^{2} < q_{\perp 2}^{2} < \cdots q_{\perp n}^{2} < Q^2
\label{eq:e3}$$ and in $x$, $z_i \ll 1$. The probability, in the leading log approximation, to emit a gluon is given by $$P \sim \frac{4 \alpha _s}{3 \pi} \frac{dz_i}{1-z_i}
\frac{dq_{\perp ,i}^2}{q_{\perp ,i}^2} 
\label{eq:e4}$$

This factor must however be modified because the ordering in  implies that no gluon is emitted with $q_\perp$ between $q_{\perp,i-1}$ and $q_{\perp,i}$. Thus we must also take into account the probability that no emission has occurred between these two values. This correction 