\boldsymbol{l}^{2} + Q^{2}\alpha_{\ell}(1 - \alpha_{\ell}) ]^{2}} = \frac{\pi}{Q^{2}\alpha_{\ell}(1 - \alpha_{\ell})}.\end{aligned}$$

After performing these integrations, one gets the scattering amplitude in transversal mode $$\begin{aligned}
(\textrm{Im}A)_{T} = - \frac{s}{3} \left( \frac{M^{2}_{H}}{\pi v}  \right) \alpha_{s}^{3} \, \alpha \sum_{q} e^{2}_{q} \left( \frac{2C_{F}}{N_{c}} \right) \int \frac{d\boldsymbol{k}^{2}}{\boldsymbol{k}^{6}} \left\{ \int_{0}^{1} \frac{[\tau^{2} + (1 - \tau)^{2}][\alpha_{\ell}^{2} + (1 - \alpha_{\ell})^{2}]\boldsymbol{k}^{2}}{\boldsymbol{k}^{2} \tau (1 - \tau) + Q^{2} \alpha_{\ell}(1 - \alpha_{\ell})} \; d\alpha_{\ell} d\tau \right\}
\label{amp-im-1}\end{aligned}$$ being $v = 246\textrm{ GeV}$ the vacuum expectation value (v.e.v.) from the Higgs mechanism. Following the same procedure, the longitudinal mode of the amplitude reads $$\begin{aligned}
(\textrm{Im}A)_{L} = \frac{4s}{3} \left( \frac{M^{2}_{H}}{\pi v}  \right) \alpha_{s}^{3} \, \alpha \sum_{q} e^{2}_{q} \left( \frac{2C_{F}}{N_{c}} \right) \int \frac{d\boldsymbol{k}^{2}}{\boldsymbol{k}^{6}} \left\{ \int_{0}^{1} \frac{[\tau(1 - \tau)][\alpha_{\ell}(1 - \alpha_{\ell})]\boldsymbol{k}^{2}}{\boldsymbol{k}^{2} \tau (1 - \tau) + Q^{2} \alpha_{\ell}(1 - \alpha_{\ell})} \; d\alpha_{\ell} d\tau \right\}.
\label{amp-im-2}\end{aligned}$$ Nonetheless, a longitudinal mode for a real photon is an unphysical property and then only the transversal one is taken into account. Integrating the Eq.(\[amp-im-1\]) over $\alpha_{\ell}$ and $\tau$, one finds $$\begin{aligned}
(\textrm{Im}A)_{T} = - \frac{4s}{9} \left( \frac{M^{2}_{H}}{\pi v}  \right) \alpha_{s}^{3} \, \alpha \sum_{q} e^{2}_{q} \left( \frac{2C_{F}}{N_{c}} \right) \int \frac{d\boldsymbol{k}^{2}}{\boldsymbol{k}^{6}} \left( 1 + \frac{24\boldsymbol{k}^{8}-226Q^{2}\boldsymbol{k}^{6}-733Q^{4}\boldsymbol{k}^{4}-670Q^{6}\boldsymbol{k}^{2}-186Q^{8}}{24Q^{8}+72Q^{6}\boldsymbol{k}^{2}+72Q^{4}\boldsymbol{k}^{4}+24Q^{2}\boldsymbol{k}^{6}} \right).
\label{amp-im-3}\end{aligned}$$

Finally, only the transversal mode is retained to compute the event rate, which is 