functions

$$\begin{aligned}
l^{\mu} &=& \alpha_{\ell} q^{\prime \mu} - \left( Q^{2} + \frac{\boldsymbol{l}^{2}}{1 - \alpha_{\ell}} \right) \frac{p^{\mu}}{s} + l^{\mu}_{\perp} \\
k^{\mu} &=& - \frac{\boldsymbol{k}^{2}}{s} q^{\prime \mu} + \left[ Q^{2} + \frac{\boldsymbol{l}^{2}}{1 - \alpha_{\ell}} + \frac{(\boldsymbol{l} + \boldsymbol{k})^{2}}{\alpha_{\ell}} \right] \frac{p^{\mu}}{s} + k^{\mu}_{\perp}.\end{aligned}$$

For completeness, one can write the momentum transfer of the process as $$\begin{aligned}
\hat{t} = (k + r)^{2} \equiv - (\boldsymbol{k} + \boldsymbol{r})^{2} \approx - 4 \boldsymbol{k}^{2} = M^{2}_{H}.\end{aligned}$$

In this proposal of Higgs boson photoproduction, the last variables to be determined are the polarization vectors of the initial and final photons. As said in the Sec. I, we apply the impact factor of the photon for $t = 0$. Thus, the polarization vectors do not depend on the $t$-variable, being its sum over transversal[^1] and longitudinal components expressed by [@forshaw-book]

$$\begin{aligned}
\epsilon^{L}_{\mu} \, \epsilon^{L *}_{\nu} &=& \frac{4Q^{2}}{s} \, \frac{p_{\mu}p_{\nu}}{s} \\
\sum \epsilon^{T}_{\mu} \, \epsilon^{T *}_{\nu} &=& - g_{\mu\nu} + \frac{4Q^{2}}{s} \, \frac{p_{\mu}p_{\nu}}{s}.\end{aligned}$$

These relations complete the set of information necessary to fully compute the imaginary part of the scattering amplitude for transversal and longitudinal modes. Nevertheless, in this kinematical regime, $\alpha$ is approximately equal to 1, and $\alpha_{k}$ can be neglected as well as the product $(k \cdot p)$ 