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Loop amplitudes in quantum field theory

Periodic Reporting for period 4 - CutLoops (Loop amplitudes in quantum field theory)

Reporting period: 2020-04-01 to 2021-08-31

Discoveries in particle physics have consistently relied on high-energy collision experiments. In colliders, new particles can be produced, and properties of known particles and their interactions can be probed in detail. At the Large Hadron Collider (LHC), the experiments ATLAS and CMS are exploring phenomena at the TeV scale through proton-proton collisions. Evidence supports the recent discovery of the Higgs boson, whose properties have just started to be studied experimentally. Searches for low-scale supersymmetry and other exotic scenarios of physics beyond the Standard Model are underway, along with a range of detailed studies of the Standard Model itself. It is thus of utmost importance to compute scattering processes to high precision.

Particle colliders involve complicated scattering configurations, where the traditional method of computing probability amplitudes by Feynman rules fails to be feasibly implementable on a reasonable time scale. Recently, a set of methods has been developed to construct these mathematical functions recursively, based on newly-discovered symmetries in quantum field theory.
The objective of the CutLoops project was to explore the mathematics of singularities in particle scattering, discover algebraic relations among them, thereby to reveal deeper structure and fundamental principles in quantum field theory, and to apply these ideas to specific particle scattering processes relevant to current collider experiments.

The project succeeded in discovering an algebraic framework involving the singularities of scattering amplitudes. Novel techniques in the computation of scattering amplitudes were implemented in new software and applied to important precision computations in electroweak observables, in quantum chromodynamics (QCD), and in Higgs studies relevant to the LHC in the presence of a large QCD background.
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.
The period of the project is now over, and all of the work described above has been substantial progress beyond the original state of the art, when multi-loop calculations were far more difficult, and even one-loop amplitudes and their cuts were not thoroughly understood. The algebraic framework was originally a vague conjecture without a proper statement. The ideas that were discovered and developed in the project are currently being employed by team members and others in their research activities in different areas, ranging from pure mathematics to computations for collider physics, as well as in new contexts such as gravitational-wave astronomy.
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