Advancing Feynman integral technology beyond polylogarithms through canonical differential equations

• Applicant: Dr. Christoph Dlapa
• Subject Area: Nuclear and Elementary Particle Physics, Quantum Mechanics, Relativity, Fields
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 555751285

Differential equations play an essential role in physics. Many intriguing processes can be accurately described if one manages to find and evaluate the integrals that constitute their solution. In high-energy physics, the evaluation of Feynman integrals and their differential equations is a major bottleneck for improving the precision of theoretical predictions, giving rise to a highly interdisciplinary and active field of research, with applications ranging from particle physics, gravitational waves, the evolution of the universe and lattice gauge theory to string theory and pure mathematics. The current most powerful analytic method for evaluating Feynman integrals is the method of canonical differential equations. Despite its success, it has so far been used mainly for the simplest type of appearing functions. However, going to higher perturbative order and including additional mass scales frequently introduces more complicated types of functions, which poses a significant challenge for improving the precision of theoretical predictions. The project will attack this challenge and advance Feynman integral technology beyond the state of the art by developing an algorithmic method for finding the canonical differential equations for these new function types. This will be achieved by working on concrete processes of high phenomenological relevance: First, the production of a top-quark pair in association with a jet (ttj) at next-to-next-to-leading order (NNLO) in quantum chromodynamics (QCD) at the Large Hadron Collider (LHC). This process is sensitive to the top-quark mass, an especially important parameter of the Standard Model of particle physics, for which improved theoretical predictions are in high demand due to the LHC's current physics programme. Second, the computation of the nonlocal-in-time contributions to the dynamics of two black holes at fourth and fifth order in Newton's coupling constant, which is essential for the case of two coalescing black holes radiating gravitational waves. This process likewise requires strong precision improvements because of the highly increased sensitivities of the planned third-generation detectors Einstein Telescope (ET) and Cosmic Explorer (CE). To reach the ambitious goal of the project, there are multiple highly challenging aspects, such as the sheer complexity of the required linear algebra or the complicated function space, that need to be overcome. They will be addressed via three work packages that develop new methods and combine them with state-of-the-art technology, as well as deep insights from the mathematics of special functions. The project will result in a publicly available computer package and advance our ability to perform computations in a variety of different fields, in particular, particle physics, gravitational wave physics, string theory and pure mathematics. As such, the project itself lies at the intersection of multiple disciplines and aims to strengthen their connection.

• DFG Programme: Research Grants