(a) triangle, (b) one-particle-reducible and (c) box diagrams[^3]. They contribute to the coefficient $C_{virt}(Q^2)$ of Eq. (\[eq:nlocxn\]), $$C_{virt}(Q^2) = 2 \Re e~\frac{\int_{\hat t_-}^{\hat t_+} d\hat t \left\{
(C_\triangle F_\triangle + F_\Box)^* [C_\triangle (\Delta
F_\triangle) + \Delta F_\Box] + G_\Box^* (\Delta G_\Box) \right\} }
{\int_{\hat t_-}^{\hat t_+} d\hat t \left\{ |C_\triangle F_\triangle +
F_\Box |^2 + |G_\Box|^2 \right\} },$$ where $\Delta F_\triangle, \Delta F_\Box$ and $\Delta G_\Box$ denote the virtual corrections to the corresponding LO form factors. While $\Delta
F_\triangle$ involves only virtual corrections to the triangle diagram, $\Delta F_\Box$ and $\Delta G_\Box$ acquire contributions from the one-particle-reducible and box diagrams.

### Triangle diagrams

![\[fg:triadia\] *Two-loop triangle diagrams contributing to Higgs-pair production via gluon fusion.*](./plots/triangles.pdf){width="90.00000%"}

The generic 2-loop triangle diagrams contributing to the virtual coefficient $C_{virt}(Q^2)$ are shown in Fig. \[fg:triadia\]. They only contribute to the spin-0 form factor $F_1$ of Eq. (\[eq:lomat\]) and can be parametrized as the correction $\Delta F_\triangle$ to the form factor $F_\triangle$, $$\Delta F_\triangle = \frac{\alpha_s}{\pi}~{\cal
C}_{virt}(Q^2)~F_\triangle \, ,$$ where ${\cal C}_{virt}(Q^2)$ denotes the [*complex*]{} virtual coefficient relative to the LO form factor $F_\triangle$ of the amplitude. This virtual coefficient is related to the single-Higgs case so that the relative QCD corrections can be simply obtained 