High-Energy Physics at the Frontier with Mathematics

Project Information

HighPHun

Grant agreement ID: 949279

DOI

10.3030/949279

EC signature date 9 November 2020

Start date 1 September 2021

End date 31 August 2026

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 502 483,00

EU contribution

€ 1 502 483,00

Coordinated by

TECHNISCHE UNIVERSITAET MUENCHEN
Germany

Periodic Reporting for period 3 - HighPHun (High-Energy Physics at the Frontier with Mathematics)

Reporting period: 2024-09-01 to 2026-02-28

Summary of the context and overall objectives of the project

The theory behind fundamental interactions among elementary particles is based on a rich mathematical framework, Quantum Field Theory (QFT). Despite more than six decades of work by physicists all over the world, many mysteries remain in the inner workings of QFT, in particular for what concerns its mathematical foundations. Unravelling these mysteries would, on the one hand, provide us with a new understanding of fundamental physics, and on the other it has the potential to provide new computational tools to address high precision calculations for particle colliders. This, in turn, will make it possible to exploit at best the enormous data sets produced by the Large Hadron Collider at CERN (LHC), and also by future colliders that will take its place, to explore fundamental interactions experimentally to the highest energy scales available.

The overall objective of this project are to advance the state of the art in theoretical particle physics, by taking a detour through pure mathematics: by building upon results on geometrical properties of complex hyper-surfaces, which are multidimensional surfaces with special mathematical properties, we are exploring new ways to obtain results for physical quantities, whose calculation till few years ago was considered out of reach due to their extreme complexity. We are developing new tools, mainly based on geometry and computer algebra, to compute previously out-of-reach processes and make precise predictions for important processes, which in turn can tell us more about how elementary particles interact and how their mass is generated through their interaction with the newly discovered Higgs boson. This has the potential to furnish a piece towards the solution of the complex puzzle that is at the basis of the origin of matter in the universe.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

During the first reporting period, the HighPHun group has mainly focussed on developing methods and technology required to achieve its long-term objectives. Some of the methods have already been applied to a multitude of important physical processes, some of which also went beyond the original goals of the action, demonstrating their importance and wide applicability in particle physics. More in detail, new ideas have been explored to handle high-precision scattering processes at particle colliders. These require manipulating huge numbers of so-called Feynman diagrams, whose complexity can elude also the larger computers available. Feynman diagrams allow to compute probabilities for particle scattering, and we have developed a new technique to handle large numbers of those and extract the relevant physical information from them efficiently. As an important result, we have used these ideas to compute the so-called three-loop amplitudes for the production of 2 jets at the Large Hadron Collider (LHC). This is a groundbreaking result on its own, since it is the first step to push the standard of precision calculations in Quantum Chromodynamics to the next level, allowing us to study to high precision how particles are produced in collisions at very high energies. In parallel, we have worked on the mathematical properties of these Feynman diagrams, exploring how general algebraic geometry and number theory can help us to compute them. In particular, we have used results on the theory of Calabi-Yau geometries, which are an important field of research in pure mathematics and found large applications in string theory, to compute Feynman diagrams relevant for collider physics. This has allowed us to compute the first complete scattering amplitude defined on an elliptic geometry, the 2 loop QED corrections to Bhabha scattering.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The HighPHun group has already pushed the state-of-the-art in theoretical particle physics in various respects. On the one hand, we have performed new groundbreaking calculations in QCD, which constitute the first essential ingredient towards understanding particle interactions at colliders at unprecedented precision. On the other, we have started to uncover profound connections between particle physics and pure mathematics, in particular algebraic geometry: more in detail, we have shown that some calculations in particle physics can be rephrased as the calculations of quantities defined on complex multidimensional surfaces called Calabi-Yau geometries. The results obtained open the way to new applications to understand the general structures lurking behind interactions among fundamental particles, and we expect to deepen this correspondence and to be able to apply these ideas to more processes till the end of the project.

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