One of the main ingredients that allow physicists to derive predictions from quantum field theory are the so-called scattering amplitudes, a set of mathematical quantities that encode the probabilities of how quantum particles interact. While the basic definition of a scattering amplitude dates back to the early days of quantum field theory, explicit computations of the so-called quantum loop corrections are still one of the major bottlenecks in theoretical particle physics. Since scattering amplitudes play such a central role in quantum physics, understanding their properties and mathematical structures is crucial in order to improve our knowledge of the fundamental laws of nature. Moreover, scattering amplitudes are not only interesting objects to study in their own right, but they are the main theoretical tool to make predictions for collider experiments like the LHC.
Despite their importance and ubiquity in modern theoretical physics, the explicit computation of scattering amplitudes is often still a bottleneck. One of the main issues when computing scattering amplitudes is the necessity to evaluate certain classes of integrals, known as Feynman integrals, which allow one to compute quantum corrections to a scattering process due to the exchange of virtual quanta. These integrals often evaluate to special functions that are poorly understood even from the mathematical side. In recent years it was realised that there are deep and far-reaching connections between scattering amplitudes and certain areas of modern mathematics, like algebraic geometry and number theory.
The goal of the project MathAm is to apply cutting-edge techniques from modern mathematics to the computation of multi-loop multi-leg scattering amplitudes in quantum field theory. Guided by advances in modern mathematics in the last decade, the goal of the project is to derive a radically new viewpoint on scattering amplitudes in quantum field theory from which the properties and the simplicity of amplitudes will be manifest at every step of the computation. The results obtained by MathAm do not only shed light on the mathematical underpinnings of quantum field theory and scattering amplitudes in general, but they also enable us to make predictions for collider experiments like the LHC at an unprecedented level of accuracy. For example, MathAm has provided the most precise predictions for the production of a Higgs boson or a pair of leptons at the LHC.
After 5 years, MathAm has been able to break new ground in a wide range of areas of physics. On the purely mathematical ground, the project has provided deep insight into the mathematical structure of Feynman integrals, and we have proposed a new algebraic structure underlying perturbative quantum field theory. The conjectures made by MathAm have received a lot of attention also from the mathematical community, and several papers have been published by mathematicians where the conjectures by MathAm have been rigorously proven. On the application side, the results of MathAm for the production of a Higgs boson are the main reference of the LHC experiments, where they play a crucial role in measuring and studying the properties of the Higgs boson, for example its couplings to matter. The results of MathAm have therefore had a very big impact on a large range of topics, and they will continue to play a crucial role in understanding of quantum field theory and the Higgs sector, and therefore fundamental laws of nature and the origin of mass.
