t_-}^{\hat t_+} d\hat t \left[
(C_\triangle F_\triangle + F_\Box)^* V_{eff}^2\right]}
{\int_{\hat t_-}^{\hat t_+} d\hat t \left\{ |C_\triangle F_\triangle +
F_\Box |^2 + |G_\Box|^2 \right\}}, \nonumber \\[0.3cm]
V_{eff} & = & F_\triangle (\bar\tau_t)\end{aligned}$$ with $\bar\tau_t = 16 m_t^2/Q^2$ \[i.e. $\tau_t$ of Eq. (\[eq:ftriangle\]) evaluated at half the invariant Higgs-pair mass $Q/2$ instead of $Q$\], where the function $F_\triangle$ can be found in Eq. (\[eq:ftriangle\]). Since Ref. [@deFlorian:2017qfk] works in the HTL, the contribution of the second form factor $F_2$ vanishes, i.e. $G_\Box\to 0$, and the approximation $V_{eff}^2/2$ is in fact treated as an approximation for the coefficient $c_1$ of the exact expression of $C_{\triangle\triangle}$ as given in Eq. (\[eq:coeffvirt\])[^5]. Thus, the approximate expression involving the coefficient $c_1$ has to be compared to the corresponding expression involving the exact coefficient $c_1$ of Eq. (\[eq:1prc1\]). This comparison is presented, normalized to the exact expression, in Fig. \[fg:1prcomp\] and shows that the approximation of Ref. [@deFlorian:2017qfk] is not better than the HTL.

### Box diagrams

The third class of two-loop contributions to the virtual corrections is given by the box diagrams. The generic box diagrams are shown in Figs. \[fg:boxdia1\]–\[fg:boxdia4\] in the Appendix. The simultaneous exchange of the gluons and Higgs bosons has to 