DL contributions when the softest particle is an electroweak boson. However, the softest particle can also be a virtual fermion. In this case, DL contributions from the integration over the momentum $k$ of the softest fermion arise from the diagram shown in Fig. 3 where the amplitudes $A_j$ are factorized into two on-shell amplitudes in the $t$-channel. We denote this contribution by $Q_j(s, \mu^2, M^2)$.

![Softest fermion contribution.](Ampfig3.epsi)

The analytic expression for $Q_j$ is rather cumbersome. However it looks simpler when the Sudakov parameterization is introduced for the softest quark momentum $k$ (with $p_1$ and $p_2$ being the initial lepton momenta). $$\label{sudak}
k = \alpha p_2 + \beta p_1 + k_{\perp}~.$$ After simplifying the spin structure, we obtain $$\label{q}
Q_j(s, \mu^2, M^2) = 
c_j \int_{\mu^2}^s 
d k_{\perp}^2 \int \frac{d \alpha}{\alpha}\frac{ d \beta}{\beta} 
\frac{2k_{\perp}^2 }{(s\alpha\beta - k_{\perp}^2)^2} 
A_j(s\alpha, k_{\perp}^2, M^2)
A_j(s\beta, k_{\perp}^2, M^2) ~,$$ where $$\label{c}
c_1 = c_2 = -c_3 = -c_4 = \frac{1}{8 \pi^2}~.$$

Similarly to Eq. (\[gtot\]), $Q_j$ of Eq. (\[q\]) can be divided into the following simple contributions: $$\label{qtot}
Q_j = Q_j^{(QED)} - \tilde{Q}_j^{(QED)} + \tilde{Q}_j + {Q'}_j ~,$$ where $$\label{qqed}
Q_j^{(QED)} (s/\mu^2)= 
c_j \int_{\mu^2}^s 
d k_{\perp}^2 \int \frac{d \alpha}{\alpha}\frac{ d \beta}{\beta} 
\frac{k_{\perp}^2 }{(s\alpha\beta - k_{\perp}^2)^2} 
A_j^{(QED)}(s\alpha/ 