\int d\alpha_{\ell} \, d\beta_{\ell} \, d^{2}\boldsymbol{l} \int d\alpha_{k} \, d\beta_{k} \, d^{2}\boldsymbol{k} \;\; \delta \left[ \beta_{\ell} + \frac{Q^{2}}{s} + \frac{\boldsymbol{l}^{2}}{s(1 - \alpha_{\ell})} \right] \delta \left[ \beta_{k} + \frac{(\boldsymbol{l} + \boldsymbol{k})^{2}}{\alpha_{\ell} s} + \beta_{\ell} \right] \, \delta [\alpha_{k} s + \boldsymbol{k}^{2}].\end{aligned}$$

Considering the Cutkosky rules to calculate the scattering amplitude, the quarks are on-mass shell due to the delta functions in Eq.(\[dps\]), which allows one to perform the following approximation to the gluon momentum $$\begin{aligned}
k^{2} \simeq - \boldsymbol{k}^{2} \hspace{2.cm} r^{2} \simeq - \boldsymbol{r}^{2} \approx - \boldsymbol{k}^{2}.\end{aligned}$$ Physically, this approximantion is an important feature, since it accesses the kinematical region of the $H \to b\bar{b}$ decay mode and experimentally expresses that the quarks are scattered in small angles [@forshaw-KMR].

The integration of the delta functions results in the following imaginary part of the amplitude $$\begin{aligned}
\textrm{Im} A = V \left( \frac{2\alpha_{s}^{2} \, \alpha}{\pi^2 s} \right) \sum_{q} e^{2}_{q} \left( \frac{\epsilon_{\mu} \epsilon^{*}_{\nu}}{N_{c}} \right) (t^{a}t^{a}) \int d\alpha_{\ell} \, \frac{d^{2}\boldsymbol{k}}{\boldsymbol{k}^{6}} \, d^{2}\boldsymbol{l} \left[ \frac{(1 - \alpha_{\ell})}{\alpha_{\ell}} \frac{T^{\mu\lambda\sigma\nu}}{(D_{1})^{2}} + \frac{T^{\lambda\mu\sigma\nu}}{D_{1}D_{2}} \right] \left[ p_{\lambda}p_{\sigma} - \frac{(k \cdot p)}{\boldsymbol{k}^{2}} p_{\lambda} r_{\sigma} \right],
\label{amp-img-fhiggs}\end{aligned}$$ with the quantities $T^{\mu\lambda\sigma\nu}$ and $T^{\lambda\mu\sigma\nu}$ being the traces present in Eq.(\[amp-cut-fhiggs\]), and the decomposed vectors acquire its coefficients from the integration of the delta 