q)$ denote the parton-parton luminosities, defined analogously to $d{\cal L}^{gg}/d\tau$ of Eq. (\[eq:lgg\]), using the quark densities $q(x,\mu_F)$, $$\begin{aligned}
\frac{d{\cal L}^{gq}}{d\tau} & = & \sum_{q,\bar q} \int_\tau^1 \frac{dx}{x}
\Big[ g(x,\mu_F) q\left(\frac{\tau}{x},\mu_F\right)
+ q(x,\mu_F) g\left(\frac{\tau}{x},\mu_F\right) \Big] \, , \nonumber \\
\frac{d{\cal L}^{q\bar q}}{d\tau} & = & \sum_q \int_\tau^1 \frac{dx}{x}
\Big[ q(x,\mu_F) \bar q\left(\frac{\tau}{x},\mu_F\right)
+ \bar q(x,\mu_F) q\left(\frac{\tau}{x},\mu_F\right) \Big]\end{aligned}$$ at the factorization scale $\mu_F$ and $P_{ij}(z)~(i,j=g,q,\bar q)$ are the specific Altarelli–Parisi splitting functions [@Altarelli:1977zs].

The quark-mass dependence is in general encoded in the LO cross section $\hat\sigma_{LO}(Q^2)$ and the terms $C_{virt}(Q^2)$, $d_{ij}(Q^2,z)$ for the virtual and real corrections, respectively. These expressions can easily be converted into the differential cross section with respect to $Q^2$, $$\begin{aligned}
\frac{d\Delta\sigma_{virt}}{dQ^2} & = & \left.
\frac{\alpha_s\left(\mu_R\right)}{\pi}~\frac{d{\cal L}^{gg}}{d\tau}~
\frac{\hat{\sigma}_{LO}\left(Q^2 \right)}{s}~C_{virt}
\left(Q^2\right) \right|_{\tau = \frac{Q^2}{s}}, \nonumber \\[0.3cm]
\frac{d\Delta\sigma_{ij}}{dQ^2} & = & \left.
\frac{\alpha_s\left(\mu_R\right)}{\pi}\int_{\frac{Q^2}{s}}^1
\frac{dz}{z^2}~\frac{d{\cal L}^{ij}}{d\tau}~
\frac{\hat{\sigma}_{LO}\left(Q^2\right)}{s}~C_{ij}(Q^2,z) \right|_{\tau
= \frac{Q^2}{zs}} \, ,
\label{eq:nlodiff}\end{aligned}$$ while the differential cross section at LO is given in Eq. (\[eq:lodiff\]).

![\[fg:gghhvirt\] *Typical two-loop triangle (left), one-particle reducible (middle) and box (right) diagrams contributing to Higgs-pair production via gluon fusion at NLO.*](./plots/dia_virt.pdf){width="95.00000%"}

Within the HTL, the Higgs coupling to gluons can be described by an effective Lagrangian [@Ellis:1975ap; @Shifman:1979eb; @Inami:1982xt; @Spira:1995rr; @Kniehl:1995tn] $${\cal L}_\mathrm{eff} = \frac{\alpha_s}{12\pi} 