The algebraic framework for amplitudes and their singularities that is mentioned above was found to have three concrete and compatible interpretations for amplitudes, in terms of their expressions as complete functions, as coefficients in the series expansions of these functions, and as the graphs of Feynman diagrams. These results are stimulating current research in mathematics as well as theoretical physics. In the course of this work, an updated understanding of the mathematical nature of these singularities was accomplished, allowing conjectured relationships to be given a firm footing and allowing new techniques to be proposed and applied.
In another strand of work done in the project, certain types of functions that arise first at two-loop order in the perturbative diagram expansion were treated in detail at an abstract level before implementing the ideas in the context of specific scattering processes. Applications to specific scattering processes can be grouped into three areas, with different specialized tools developed for each one. One area is the study of four-loop calculations of form factors and cusp anomalous dimensions in QCD, which is a significant testing ground since the features at this order indicate the extent to which certain all-order conjectures may hold. Another area is that of two-loop electroweak effects for precision tests of the Standard Model, starting with data from the Large Electron-Positron Collider at CERN and used to inform proposals for possible future collider experiments. A third area is that of two-loop calculations that can be applied separately in the production of the Higgs boson together with one hadronic jet or in the decay of the Higgs to three partons. The project has delivered concrete results in each of these areas, going well beyond what would have been available when the project started. The techniques developed in the project are being used in current research by a wider range of scientists for other physical applications.
At the time of writing, the project has resulted in 33 scientific publications that are freely available online, and the work has been presented at numerous international conferences, seminars, and workshops. It has led to two public software packages for amplitude computations and has hosted an international conference with a public outreach component. Junior researchers have been nurtured and are continuing their careers within both the academic and private sectors.
