The LO matrix element of $g(q_1) g(q_2) \to H(p_1) H(p_2)$ can be cast into the form $$\begin{aligned}
{\cal M}(g^a g^b \to HH) & = & -i\,\frac{G_F\alpha_s(\mu_R) Q^2}{2\sqrt{2}\pi}
{\cal A}^{\mu\nu} \epsilon_{1\mu} \epsilon_{2\nu} \delta_{ab}
\nonumber \\[0.3cm]
\mbox{with} \qquad {\cal A}^{\mu\nu} & = & F_1 T_1^{\mu\nu} + F_2 T_2^{\mu\nu}
\, , \nonumber \\[0.3cm]
F_1 & = & C_\triangle F_\triangle + F_\Box \, , \qquad
\qquad \qquad
F_2 = G_\Box \, , \nonumber \\[0.3cm]
C_\triangle & = & \frac{\lambda_{H^3} v}{Q^2 - M_H^2 +
iM_H\Gamma_H} \nonumber \\[0.3cm]
\mbox{and} \qquad Q^2 & = & (p_1+p_2)^2 = m_{HH}^2
\label{eq:lomat}\end{aligned}$$ with $Q=m_{HH}$ denoting the invariant Higgs-pair mass. Here $a,b$ denote the color indices of the initial gluons, $\epsilon_{1/2}$ their polarization vectors, $\Gamma_H$ the total Higgs width[^2], $G_F$ the Fermi constant and $\alpha_s(\mu_R)$ the strong coupling at the renormalization scale $\mu_R$. Since in this work we neglect the small bottom-quark contribution, the LO function of the triangle-diagram contribution is given by the top-quark contribution, $$F_\triangle (\tau_t) = \tau_t \Big[ 1 + (1-\tau_t) f(\tau_t) \Big ]
\label{eq:ftriangle}$$ with $\tau_t = 4m_t^2/Q^2$ and the basic function $$\begin{aligned}
f(\tau) & = & \left\{ \begin{array}{ll} \displaystyle \arcsin^2
\frac{1}{\sqrt{\tau}} & \tau \ge 1 \\ \displaystyle - \frac{1}{4} \left[
\log \frac{1+\sqrt{1-\tau}} {1-\sqrt{1-\tau}} - i\pi \right]^2 