Exact two-loop amplitudes for Higgs plus jet production with a cubic Higgs self-coupling

We compute the corrections to the two-loop amplitudes for $gg\to h$, $gg\to hg$, $qg\to hq$, and $q\bar{q}\to hg$ due to a modified cubic Higgs self-coupling. The exact dependence on the Higgs and top-quark masses across the entire $2\to 2$ phase space is determined by numerically solving a system of differential equations for the relevant master integrals. The calculated amplitudes are crucial for evaluating the impact of the considered corrections on exclusive $pp\to hj$ production at the hadronic event level. As an application, we calculate the non-universal, kinematic-dependent cubic Higgs self-coupling corrections to the Higgs-boson transverse momentum distribution in gluon-gluon fusion Higgs production for arbitrary values of the transverse momentum.