$M^2 \ll k_{\perp}^2 \ll s$. This yields: $$\label{gwz}
G_j^{(WZ)} = - \frac{1}{8\pi^2} b_j^{(WZ)}
\int_{M^2}^s \frac{dk_{\perp}^2 }{k_{\perp}^2} 
\ln(s/k_{\perp}^2)\tilde{A}_j(s/k_{\perp}^2 ) ~,$$ with $$\label{bwz}
b_j^{(WZ)} = b_j - b_j^{\gamma} ~$$ and the factors $b_j$ can be taken from Ref. [@egt]: $$\begin{aligned}
\label{btot}
b_1 = \frac{{g'}^2 (Y_1 - Y_2)^2}{4}, 
~~b_2 = \frac{8g^2+{g'}^2 (Y_1 - Y_2)^2}{4}, \\ \nonumber
~~b_3=\frac{{g'}^2 (Y_1 + Y_2)^2}{4},
~~b_4 = \frac{8g^2 + {g'}^2 (Y_1 + Y_2)^2}{4}~~.\end{aligned}$$

In Eq. (\[gwz\]) we have used the fact that the $W$ and the $Z$ bosons cannot be the softest particles for the amplitudes $A^{(QED)}_j$ since the integrations over the softest transverse momenta in $A^{(QED)}_j$ can go down to $\mu$, by definition. The sum of Eqs. (\[gphot\]) and (\[gwz\]) , $G_j$ can be written in the more convenient way: $$\label{gtot}
G_j(s, \mu^2, M^2) = 
G^{(\gamma)}_j(s, \mu^2, M^2)+ G_j^{(WZ)}(s,M^2) =
G_j^{(QED)}(s, \mu^2) - 
\tilde{G}^{(QED)}_j(s, M^2) + \tilde{G}_j(s, M^2) + {G'}_j(s, \mu^2, M^2) ~,$$ where $$\begin{aligned}
\label{g}
G^{(QED)}_j = - \frac{1}{8\pi^2} b_j^{(\gamma)} 
\int_{\mu^2}^s \frac{dk_{\perp}^2 }{k_{\perp}^2} 
\ln(s/k_{\perp}^2)A^{(QED)}_j(s/k_{\perp}^2) ~,
\tilde{G}^{(QED)}_j = -\frac{1}{8\pi^2} b_j^{(\gamma)} 
\int_{M^2}^s \frac{dk_{\perp}^2 }{k_{\perp}^2} 
\ln(s/k_{\perp}^2)\tilde{A}^{(QED)}_j (s/k_{\perp}^2) ~, \\ \nonumber 
\tilde{G}_j = - \frac{1}{8\pi^2} b_j \int_{M^2}^s
\frac{dk_{\perp}^2 }{k_{\perp}^2} 
\ln(s/k_{\perp}^2)\tilde{A}_j (s/k_{\perp}^2) ~,
{G'}_j = -\frac{1}{8\pi^2} b_j^{(\gamma)} 
\int_{\mu^2}^{M^2} \frac{dk_{\perp}^2 }{k_{\perp}^2} 
\ln(s/k_{\perp}^2){A'}_j (s/k_{\perp}^2, M^2/k_{\perp}^2 ) ~. \end{aligned}$$

Eqs. (\[gtot\], \[g\]) account for 