Hypergeometric Functions and Feynman Diagrams

2020/12/28 by Kalmykov, Mikhail, Bytev, Vladimir, Kniehl, Bernd +3 · 3 citations
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Phenomenology (hep-ph) #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2012.14492

Abstract

The relationship between Feynman diagrams and hypergeometric functions is discussed. Special attention is devoted to existing techniques for the construction of the ε-expansion. As an example, we present a detailed discussion of the construction of the epsilon-expansion of the Appell function F₃ around rational values of parameters via an iterative solution of differential equations. As a by-product, we have found that the one-loop massless pentagon diagram in dimension d=3-2ε is not expressible in terms of multiple polylogarithms. Another interesting example is the Puiseux-type solution involving a differential operator generated by a hypergeometric function of three variables. The holonomic properties of the F_N hypergeometric functions are briefly discussed.

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