$A_j,~(j = 1,2,3,4)$ with respect to an infrared cut-off. We introduce two such cut-offs, $\mu$ and $M$. We presume that $M \approx M_Z \approx M_W$ and use this cut-off to regulate the DL contributions involving soft (almost on-shell) virtual $W,Z$ -bosons. In order to regulate the IR divergences arising from soft photons we use the cut-off $\mu$ and we assume that $\mu \approx m_q \ll M$ where $m_q$ is the maximal quark mass involved. Both cut-offs are introduced in the transverse momentum space (with respect to the plane formed by momenta of the initial leptons) so that the transverse momenta $k_i$ of virtual photons obey $$\label{mu}
k_{i\perp} > \mu ~,$$ while the momenta $k_i$ of virtual $W,Z$ -bosons obey $$\label{M}
k_{i\perp} > M ~.$$

Let us first consider $A_j$ in the collinear kinematics where, in the cm frame, the produced quarks or leptons move very close to the $e^+e^-$ -beams. In order to fix such kinematics, we implement Eq. (\[tkin\]) by the further restriction on $t$: $$\label{tmu}
s \sim -u \gg M^2 \gg \mu^2 \geq -t$$ and similarly for Eq. (\[ukin\]) by $$\label{umu}
s \sim -t \gg M^2 \gg \mu^2 \geq -u ~.$$

Basically in DLA, the invariant amplitudes $A_j$ 