Feynman integrals and hyperlogarithms

Abstract

We study Feynman integrals in the representation with Schwinger parameters and derive recursive integral formulas for massless 3- and 4-point functions. Properties of analytic (including dimensional) regularization are summarized and we prove that in the Euclidean region, each Feynman integral can be written as a linear combination of convergent Feynman integrals. This means that one can choose a basis of convergent master integrals and need not evaluate any divergent Feynman graph directly. Secondly we give a self-contained account of hyperlogarithms and explain in detail the algorithms needed for their application to the evaluation of multivariate integrals. We define a new method to track singularities of such integrals and present a computer program that implements the integration method. As our main result, we prove the existence of infinite families of massless 3- and 4-point graphs (including the ladder box graphs with arbitrary loop number and their minors) whose Feynman integrals can be expressed in terms of multiple polylogarithms, to all orders in the epsilon-expansion. These integrals can be computed effectively with the presented program. We include interesting examples of explicit results for Feynman integrals with up to 6 loops. In particular we present the first exactly computed counterterm in massless phi^4 theory which is not a multiple zeta value, but a linear combination of multiple polylogarithms at primitive sixth roots of unity (and divided by $\sqrt{3}$). To this end we derive a parity result on the reducibility of the real- and imaginary parts of such numbers into products and terms of lower depth.

PhD thesis, 220 pages, many figures

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Cited by in corpus (55)

• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
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• Numbers and Functions in Quantum Field Theory
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• Toric geometry and regularization of Feynman integrals
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• Relativistic causality and position space renormalization
• Algebraic differential formulas for the shuffle, stuffle and duality relations of iterated integrals
• Multi-valued Feynman Graphs and Scattering Theory
• Generalizations of polylogarithms for Feynman integrals
• Duality/Sum formulas for iterated integrals and their application to multiple zeta values
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• The signed permutation group on Feynman graphs
• Hyperlogarithmic functional equations on del Pezzo surfaces
• The motivic Galois group of mixed Tate motives over $\mathbb{Z}[1/2]$ and its action on the fundamental group of ${\mathbb{P}}^1\setminus \{0,\pm 1,\infty \}$