\leftrightarrow \hat u) \right\}}
{\int_{\hat t_-}^{\hat t_+} d\hat t \left\{ |C_\triangle F_\triangle +
F_\Box |^2 + |G_\Box|^2 \right\}}, \nonumber \\[0.3cm]
C^\infty_{\triangle\triangle} & = & \left. C_{\triangle\triangle}
\right|_{c_1 = 2/9}, \nonumber \\[0.3cm]
d_{gg}(Q^2,z) \!\!\! & \to & \!\!\! - \frac{11}{2} (1-z)^3 \, , \
d_{gq}(Q^2,z) \to \frac{2}{3} z^2 - (1-z)^2 \, , \
d_{q\bar q}(Q^2,z) \to \frac{32}{27} (1-z)^3 \, ,
\label{eq:coeffvirt}\end{aligned}$$ where $\hat s,\hat t, \hat u$ ($\hat s=Q^2$ at LO and for the virtual corrections) denote the partonic Mandelstam variables and $C_{\triangle\triangle}$ is the contribution of the one-particle reducible diagrams, see Fig. \[fg:gghhvirt\].

At NLO QCD, the full mass dependence of the LO partonic cross section has been taken into account, while keeping the virtual corrections $C_{virt}$ and the real corrections $d_{ij}$ in the HTL (“Born-improved” approach) [@Dawson:1998py]. This yields a reasonable approximation for smaller invariant Higgs-pair masses and approximates the full NLO result of the total cross section within about 15% [@Borowka:2016ehy; @Borowka:2016ypz; @Baglio:2018lrj]. The NLO QCD corrections in the HTL increase the cross section by $80-90\%$ [@Dawson:1998py]. Within the Born-improved HTL, the NNLO QCD corrections have been obtained in Refs. [@deFlorian:2013uza; @deFlorian:2013jea; @Grigo:2014jma] increasing the total cross section by a moderate amount of $20-30\%$ [@deFlorian:2013jea]. Beyond these NNLO 