A two-loop self-dual Higgs plus four gluon amplitude: the all-plus configuration

QCD corrections at two-loop order for the Higgs plus four gluon amplitude are fundamental ingredients to obtain the theoretical predictions at NNLO for the Higgs production with two jets. In order to capture these QCD effects, we can work in the large top mass limit and treat the Higgs as the real part of a complex scalar field (phi) coupled to the self-dual part of the gluon field strength. In this project, we apply on-shell techniques to compute the discontinuities of a two-loop phi plus four gluon amplitude which represents one of the two contributions to the corresponding Higgs amplitude. Firstly, we review some on-shell methods to compute scattering amplitudes. We introduce the spinor-helicity formalism to ensure on-shell simplicity is manifest and summarise methods of recursion relations at tree-level and unitarity techniques at loop level. The interaction between the phi field and gluons presents a vanishing tree-level amplitude in the all-plus sector, as observed in gauge theories. Led by this analogy, we start by understanding the structure of one-loop and two-loop amplitudes in the all-plus sector of Yang-Mills. Then, after a summary of some known results about the interaction between the self-dual Higgs and all-plus gluons at tree and one-loop level, we describe our computation of the cut-constructible pieces for the two-loop amplitude which couples phi to four gluons in the all-plus configuration. After subtracting terms related to the universal pole structure, we obtain the finite remainder function written in terms of logarithms and dilogarithms. Finally, we check the factorisation properties in collinear limits.