Computer Algebra for Multi-Loop Feynman Integrals

In the digital age many areas of mathematics have been algorithmized. Not only simple calculators but often sophisticated apps for the manipulation of non-trivial formulas are installed on many smartphones that are used naturally in many areas of life, like in school, university studies or in business. But this is only the tip of the iceberg. With advanced computer algebra technologies it is meanwhile possible to simplify complex expressions that cannot be treated anymore by experts because of their complexity or involved size. In this research project in close cooperation with the Deutsches Elektronen-Synchrotron (DESY, Johannes Blümlein und Peter Marquard) we will apply these exciting developments of algorithmic mathematics to the research area of particle physics. In order to answer or at least approach fundamental questions such as "What did the universe look like in the first second after the big bang?", "Do the 4 fundamental forces unite at high energies?", or "Do the properties of the new particle, called the Higgs-Boson comply with the theoretical predictions?", highly complicated Feynman integrals have to be simplified that describe the interaction of elementary particles with high precision. More precisely, we will simplify challenging expressions of massive multi-loop Feynman integrals that require several GBs of memory. Here the basic idea is to rewrite the expressions in a preprocessing step to alternative expressions that fit into the input class of our computer algebra algorithms. Besides the transformation to more suitable integral representations, it will be necessary to produce nested sums or coupled linear differential equations (linear systems with one extra differential operator) that contain the physical problems as solutions. In all these cases it will be a central task to solve gigantic linear differential equations and recurrences where the solutions are composed also in terms of new special functions whose mathematical properties are completely unknown. In order to carry out these monster calculations, many non-trivial obstacles have to be overcome that will be only feasible with a new generation of computer algebra technologies. In particular, new algorithms within the field of symbolic summation and integration but also new technologies from the field of special functions will be developed. A special challenge will be the implementation of these sophisticated algorithms in form of stable and highly efficient software packages. In summary, we plan to simplify gigantic expressions of multi-loop Feynman integrals with the help of our optimized and novel computer algebra technologies and will process them further to a form that is urgently needed, e.g., for the Large Hardron Collider (LHC) and its planned successor at CERN, the FCC. In particular, our results will contribute substantially to gain further fascinating insight within the world of particle physics.

To understand the fundamental building blocks of the microcosm, scientists perform extremely high-energy collisions at CERN's Large Hadron Collider (LHC). In these experiments, as well as at future accelerators, measurement accuracy is crucial: to detect even the smallest deviations that could point to previously unknown laws of nature and new elementary particles, theoretical predictions in physics must be far more precise than the experimental data. This is precisely where our project comes in. The goal was to calculate extremely complex mathematical processes of quantum chromodynamics (the theory of strong interactions) with exact precision. At the center were so-called 3-loop calculations - highly complex quantum corrections that are essential for accurately determining the production rate of top quarks (the heaviest known elementary particle), as well as other heavy quarks. Since manual calculations of these processes far exceed human capabilities, the project team, in collaboration with the German Electron Synchrotron (DESY), developed novel methods in computer algebra and the theory of higher special functions. These software tools make it possible to fully automate the simplification of gigantic expressions of Feynman integrals and evaluate them numerically with extremely high precision. The key milestones of the project: Breakthrough in 3-loop form factors: After years of preparatory work, the team succeeded in decoding the enormously difficult mathematical contributions to the top-anti-top quark production rate at the LHC and at future electron-positron colliders, such as the planned FCC-ee at CERN. By calculating up to 40,000 series coefficients and utilizing specialized computer algebra software, physical expressions could be determined with a precision of up to 4,000 decimal places and translated into exact mathematical formulas. Completion of an 18-year long-term project: The calculation of the so-called massive 3-loop operator matrix elements was fully completed in collaboration between the Research Institute for Symbolic Computation (RISC) and DESY. For the first time, these results provide the global physics community with fast, high-precision tools to correctly describe the internal structure of protons in collisions at the 3-loop level, which will also impact future extractions of the value of the strong coupling constant. In addition to the physical results, the project led to significant advances in symbolic mathematics and the theory of special functions. The algorithms developed for solving coupled differential and difference equations, analyzing and determining the properties of the special functions that arise, and simplifying complex sums are now even finding applications in pure number theory.