The 5th International Congress on Mathematical Software
Abstract: We consider zero and single scale higher loop Feynman integrals in renormalizable quantum field theories in four space-time dimensions. Their calculation reveals, from low to higher orders, a sequence of special number- and function-spaces. We review the present status and give a detailed characteristics of these spaces, w.r.t. their properties and representations. These objects are of key relevance of all precision calculations for observables measured in high energy scattering experiments of elementary particles.
Abstract: In this talk I present the Maple program MPL for symbolic computations with multiple polylogarithms. This class of functions, which can be understood as a very general, multi-variable generalization of the logarithm function, plays an important role in quantum field theory and in algebraic geometry. The program MPL views these functions as iterated integrals and allows for various operations, such as differentiations, the construction of bases of corresponding vecto spaces, the evaluation of certain limits and of definite integrals over these functions. I will discuss possible applications to the computation of certain period integrals and Feynman integrals.
Abstract: We report on the status of our ongoing efforts to calculate the 3-loop heavy flavor contributions to Wilson coefficients and structure functions in DIS.
Abstract: Recent progress in computational techniques for high energy physics connects high energy physics with abstract mathematics. Notions from algebraic and arithmetic geometry play a crucial role for the practitioner of Feynman diagram computation. We look at examples.
Abstract: In Quantum Field Theory, calculations of multi-loop scattering amplitudes often rely on integration-by-parts reductions of dimensionally regularised Feynman integrals. In this talk, I show how modular image and reconstruction techniques can help to speed-up the computation of such reduction identities.
Abstract: Nested multiple sums of monomials are one way to represent multiple polylogarithms (MPL). These functions are well-known and supported by several computer algebra programs. In many physical applications, however, they can appear initially in much more complicated disguise. Expressing a given multiple sum in terms of a basis of MPL, and deciding whether this is possible at all, are therefore important open problems. In this talk I consider the class of conical sums. These are generalizations of the sums defining MPL in two ways: On one hand, the summand may contain powers of arbitrary linear forms. On the other hand, the summation domain is the set of lattice points in an arbitrary cone. I will explain how these sums can be transformed into multiple polylogarithms, using symbolic integration and convex geometry.
Abstract: An outline of the possibility to express Feynman diagrams in terms of special functions defined through integral representations is given. By switching to summation definitions of special functions a problem of summation is posed. Refined holonomic summation techniques are introduced as a possibility to solve such problems. The relative advantages and disadvantages to other techniques is highlighted with examples.
Abstract: Symbolic summation started with Abramov's telescoping algorithm for rational functions (1971), was pushed further by Gosper's algorithm for hypergeometric expressions (1978) and reached its first peak level with Zeilberger's creative telescoping algorithm (1990) to treat definite hypergeometric sums. Many further developments have advanced this field in the recent years in order to handle such summation inputs and generalizations of them in a very efficient way. In this talk we focus on the difference ring approach whose foundation was lead by Karr's summation algorithm (1981). In the last 15 this approach has been enlarged significantly to a refined difference field and new difference ring theory and has been incorporated within the summation package Sigma. In particular, being faced with extremely challenging problems from QCD we were driven to optimize Sigma and to develop the complementary packages EvaluateMultiSums, SumProduction and SolveCoupledSystem. In this way we can simplify up to seven-fold sums automatically, and we are able to crunch large expressions and to solve huge coupled systems of several GBs. We will illustrate this software machinery by concrete examples arising from QCD.
Abstract: Feynman integrals are usually computed within dimensional regularisation. A given Feynman integral evaluates to a Laurent series in the dimensional regularisation parameter. In this talk I will discuss algorithms, which allow systematically to compute the coefficients of the Laurent series. I will review the case, where the coefficients can be expressed in terms of multiple polylogarithms and present new results involving generalisations towards the elliptic case.