Symbolic Summation in Perturbative Quantum Field Theory

The calculation of Feynman parameter integrals plays an important role in perturbative quantum field theory. In higher order perturbation theory the analytic evaluation of the Feynman parameter integrals is not solved in general. Depending on the mass-scale parameters of the problem and the number of external invariants, these multiple integrals form classes of functions which are generalized hypergeometric functions and generalizations thereof. The current methods in particle physics are instrumental to match the experimental accuracy with which the fundamental parameters of the Standard Theory of the Elementary Particles are measured. Thorough progress in this field can only be achieved by applying innovative mathematical technologies, which requires the close collaboration of mathematicians and theoretical physicists, as anticipated in the present proposal. Only in this way fundamental questions as: "How and where do the fundamental forces in our universe unify?" can be answered. A precise calculation within the present Standard Theory, being possible only in this way, will allow to reveal also very small signs of new physics and therefore make highest use out of the very cost-intensive accelerator and detector facilities. The calculation of these integrals requires efficient and rather flexible symbolic summation techniques, as well as special function algorithms that assist in this task, including simplification and verification algorithms. We will develop and explore new algorithmic approaches to handle such classical and new classes of special functions. In course of calculating new classes of Feynman diagrams for the first time, new general classes of mathematical functions will emerge and their properties will be derived. As experienced in the past, classes of this type do soon find applications in many branches of science and technology. We will accelerate this process in providing related computer algebra algorithms and corresponding software implementations; in addition we will document results on new function classes through publications and web-sites. Our proposed research will be carried out in an intensive collaboration with the Deutsches Elektronen-Synchrotron (DESY), Member of the Helmholtz Association, which is one of the leading accelerator centers and centers for elementary particle research in the world. From the proposed project we expect a synergistic effect that should open up new avenues of research, both in particle physics and in symbolic computation and special functions.

A central goal of the project was the exploration of elementary particles and the fundamental forces, in particular the strong force, that act between them. The latter one holds together quarks and gluons within protons and nucleons of the atoms. Its size is less well determined than the electromagnetic force transmitted by the light quarks and the weak force which is responsible for the natural radioactivity. The interaction of the quarks and gluons is described by the so-called Feynman diagrams, resp. Feynman integrals. The symbolic evaluation and simplification of those integrals is the crucial step in order to derive physical statements about the behaviour of elementary particle and their forces. In this project new algorithms and technologies for symbolic summation, symbolic integration and special functions have been developed to push forward challenging calculation of new classes of Feynman integrals that are urgently needed in particle physics. We provided a systematic method to transform the Feynman diagrams under consideration in terms of convergent nested sum representations containing both finite and infinite sums over hypergeometric terms. The resulting summations (up to seven fold sums!) and the summation objects were highly intertwined. The essential task was then to uncouple these sum expressions to simpler sums for further processing. In order to tackle such highly challenging sums, the available summation theory and its summation algorithms needed to be improved substantially. In this way, we succeeded, e.g., in solving voluminous recurrence relations. Finally, in order to solve the given physical problem in terms for Feynman integrals, we were faced with about 1 million of such complicated sums. Thus new summation tools and packages needed to be developed and implemented in order to deal with such large scale problems. Meanwhile we arrived at a completely automatic summation toolbox that has been executed on large capacity computers with 200 Gbyte RAM. While performing the evaluation of the new classes of Feynman integrals, we started to leave the class of known functions in which the simplification of Feynman integrals can be given and entered more general classes of functions that are not well understood. To facilitate all these calculations and to produce compact output, algebraic and analytic properties of these new function classes have been explored and algorithms were developed to calculate the needed properties efficiently.This project has been carried out with the cooperation partner DESY (German Electron-Synchrotron), a leading institute in the exploration of particle physics. The derived and ongoing calculations based on the projects computer algebra technologies will be used, e.g., by the precision measurements of the Hadron Electron Accelerator (HERA) of DESY and of the Large Hadron Collider (LHC) of CERN.