Periodic Reporting for period 1 - LoCoMotive (Loop Corrections from the Theory of Motives)
Reporting period: 2023-01-01 to 2025-06-30
The cornerstone to make theoretical predictions for many particle physics and gravitational wave experiments are so-called scattering amplitudes, which are mathematical quantities that describe the way, e.g. fundamental particles interact. The computation of scattering amplitudes is notoriously difficult, which is one of the major bottlenecks when computing theoretical predictions. Over the last decade it was realised that there are deep connections between the computation of scattering amplitudes and certain branches of modern mathematics, in particular algebraic geometry and number theory.
The goals of the project LoCoMotive is to develop new technique beyond the state of the art for the computation of scattering amplitudes based on novel ideas from modern mathematics. In order to achieve its goals, the activities of LoCoMotive are divided into 3 main areas. First, we will perform in in-depth study of the areas of mathematics needed to describe scattering amplitudes, and will develop the relevant mathematics further where needed. Second, we aim at studying scattering amplitudes in a variety of physics theories which exhibit additional symmetries, in order to identify new structures and results in these theories. Finally, we aim at performing very precise predictions for experiments that are out of reach with conventional techniques, in particular collider experiments like the Large Hadron Collider (LHC) at CERN and its possible successors.
The goals of the project LoCoMotive is to develop new technique beyond the state of the art for the computation of scattering amplitudes based on novel ideas from modern mathematics. In order to achieve its goals, the activities of LoCoMotive are divided into 3 main areas. First, we will perform in in-depth study of the areas of mathematics needed to describe scattering amplitudes, and will develop the relevant mathematics further where needed. Second, we aim at studying scattering amplitudes in a variety of physics theories which exhibit additional symmetries, in order to identify new structures and results in these theories. Finally, we aim at performing very precise predictions for experiments that are out of reach with conventional techniques, in particular collider experiments like the Large Hadron Collider (LHC) at CERN and its possible successors.
In a first strand of research, Duhr and Porkert and collaborators have analysed mathematical properties of multi-loop Feynman integrals. Using tools from twisted cohomology theory, we could show that all Feynman integrals in 2 dimensions with generic propagator powers can be expressed in terms of single-valued hypergeometric functions. We also used techniques from twisted cohomology to study non-linear relations between Feynman integrals and their cuts, and to understand a notion of self-duality for maximal cuts.
Duhr and Porkert and collaborators have used techniques from Calabi-Yau geometry to compute new results for various Yangian-invariant traintrack integrals in two space-time dimensions.
Zhang and collaborators have bootstrapped new results for two-loop integrals that evaluate to non-polylogarithmic functions.
We have computed novel results for various integrals of phenomenological relevance involving massive particles. In particular, Duhr and collaborators have computed the complete two-loop corrections to Bhabha scattering in Quantum Electrodynamics (QED) with the full dependence on the electron mass. This computation has been an open problem for more than a decade. Together with his collaborators, he has also computed for the first time the three-loop QED corrections to the electron self-energy. Chaubey and collaborators have computed for the first time integrals relevant for light-by0light scattering, and in collaboration with Maggio, she has computed a highly non-trivial non-planar family of four-point functions depending on a massive propagator.
Duhr and Porkert and collaborators have used techniques from Calabi-Yau geometry to compute new results for various Yangian-invariant traintrack integrals in two space-time dimensions.
Zhang and collaborators have bootstrapped new results for two-loop integrals that evaluate to non-polylogarithmic functions.
We have computed novel results for various integrals of phenomenological relevance involving massive particles. In particular, Duhr and collaborators have computed the complete two-loop corrections to Bhabha scattering in Quantum Electrodynamics (QED) with the full dependence on the electron mass. This computation has been an open problem for more than a decade. Together with his collaborators, he has also computed for the first time the three-loop QED corrections to the electron self-energy. Chaubey and collaborators have computed for the first time integrals relevant for light-by0light scattering, and in collaboration with Maggio, she has computed a highly non-trivial non-planar family of four-point functions depending on a massive propagator.
The computation of processes involving massive particles may serve as an input to provide very precise predictions for collider observables. In particular, the two-loop corrections to Bhabha scattering may be used to provide very precise predictions for current and future electron-position colliders.