G^{a\mu\nu} G^a _{\mu\nu}
\left(C_1 \frac{H}{v} - C_2 \frac{H^2}{2v^2} \right)
\label{eq:leff}$$ involving the Wilson coefficients ($L_t = \log \mu_R^2/m_t^2$) [@Chetyrkin:1997iv; @Kramer:1996iq; @Dawson:1998py; @Schroder:2005hy; @Baikov:2016tgj; @Grigo:2014jma; @Spira:2016zna; @Gerlach:2018hen] $$\begin{aligned}
C_1 & = & 1 + \frac{11}{4} \frac{\alpha_s}{\pi} + \left\{
\frac{2777}{288} +
\frac{19}{16} L_t + N_F \left(\frac{L_t}{3}-\frac{67}{96} \right)
\right\} \left(\frac{\alpha_s}{\pi} \right)^2 + {\cal O}(\alpha_s^3)
\, , \nonumber \\
C_2 & = & C_1 + \left( \frac{35}{24} + \frac{2}{3} N_F \right)
\left(\frac{\alpha_s}{\pi} \right)^2 + {\cal O}(\alpha_s^3)
\label{eq:leffcoeff}\end{aligned}$$ that are known up to N$^4$LO [@Schroder:2005hy; @Baikov:2016tgj; @Spira:2016zna]. Since the top quark is integrated out, the number of active flavours has been chosen as $N_F=5$. If these effective Higgs couplings to gluons in the calculation of the NLO QCD corrections are used, the calculation of these is simplified to a one-loop calculation for the virtual corrections and a tree-level one for the matrix elements of the real corrections. The terms $C_{virt}(Q^2)$ and $d_{ij}(Q^2,z)$, for the virtual and real corrections, approach in the HTL the simple expressions $$\begin{aligned}
C_{virt}(Q^2) & \to & \frac{11}{2} + \pi^2 +
C^\infty_{\triangle\triangle} +
\frac{33-2N_F}{6} \log\frac{\mu_R^2}{Q^2}, \nonumber \\
 C_{\triangle\triangle} & = &
\Re e~\frac{\int_{\hat t_-}^{\hat t_+} d\hat t \left\{ c_1 \left[
(C_\triangle F_\triangle + F_\Box) + \frac{p_T^2}{\hat t} 
G_\Box \right]^* + (\hat t 