q^{2}$ is the photon virtuality. Thus, the decomposed four-momenta can be written as

$$\begin{aligned}
\ell^{\mu} &=& \alpha_{\ell} q^{\prime \mu} + \beta_{\ell} p^{\mu} + \ell^{\mu}_{\perp} \\
k^{\mu} &=& \alpha_{k}q^{\prime \mu} + \beta_{k} p^{\mu} + k^{\mu}_{\perp} \\
r^{\mu} &=& \alpha_{r} q^{\prime \mu} + \beta_{r} p^{\mu} + r^{\mu}_{\perp}.\end{aligned}$$

This set of decomposed four-vectors enables one to rewrite the denominators under the traces in Eq.(\[amp-cut-fhiggs\]) as

$$\begin{aligned}
\l^{2} = - \left[ \frac{\alpha_{\ell}(1 - \alpha_{\ell})Q^{2} + \boldsymbol{l}^{2}}{1 - \alpha_{\ell}} \right] &\equiv& - \frac{D_{1}}{1 - \alpha_{\ell}} \\
(l + k - q)^{2} = - \left[ \frac{\alpha_{\ell}(1 - \alpha_{\ell})Q^{2} + (\boldsymbol{l} + \boldsymbol{k})^{2}}{\alpha_{\ell}} \right] &\equiv& - \frac{D_{2}}{\alpha_{\ell}}.\end{aligned}$$

\[d1d2\]

The final step is to write out the volume element of the three-body phase space under the Sudakov parametrization in order to obtain the imaginary part of the scattering amplitude. The definition of the volume element reads $$\begin{aligned}
 \nonumber
\int d(PS)_{3} &=& \int \frac{d^{4}f_{1}}{(2\pi)^{3}} \, \frac{d^{4}f_{2}}{(2\pi)^{3}} \, \frac{d^{4}f_{3}}{(2\pi)^{3}} \, \delta(f^{2}_{1}) \, \delta(f^{2}_{2}) \, \delta(f^{2}_{3}) \, (2 \pi)^{4} \, \delta^{4}(q + p - f_{1} - f_{2} - f_{3}) \\
&=& \frac{1}{(2 \pi)^{5}} \int d^{4}l \, d^{4}k \, \delta([q - \ell]^{2}) \, \delta([\ell + k]^{2}) \, \delta([p - k]^{2}),
\label{dps}\end{aligned}$$ which expressed under the Sudakov parametrization reads $$\begin{aligned}
\int d(PS)_{3} = 