Two-loop form factors for Dark Matter annihilation to colored Standard Model particles

We investigate a UV-complete model in which the dark matter (DM) particle interacts with gluons through a colored scalar mediator. This framework provides a phenomenologically viable scenario testable at hadron colliders. While mono-jet signatures are relevant for collider searches, zero-jet processes correspond to complete annihilation of Standard Model (SM) particles into DM, contributing to the relic density. In this work, we study dark matter annihilation into SM colored particles, which in our model arises at leading order from loop-induced processes. We compute the relevant two-loop QCD amplitudes for both gluon and quark channels in dark matter production or annihilation. The amplitudes are decomposed into scalar form factors using the projector technique. Using integration-by-parts (IBP) identities, we obtain analytical expressions for the form factors in terms of master integrals. Ultraviolet divergences are removed via counterterm renormalization, yielding UV-finite results. These results would enable predictions for dark matter production or annihilation into SM colored particles at next-to-leading order (NLO) in QCD.