Towards all-order Laurent expansion of generalised hypergeometric functions about rational values of parameters

Abstract

Hypergeometric functions provide a useful representation of Feynman diagrams occuring in precision phenomenology. In dimensional regularization, the ε-expansion of these functions about d = 4 is required. We discuss the current status of differential reduction algorithms. As an illustration, we consider the construction of the all-order ε-expansion of the Appell hypergeometric function F 1 around integer values of the parameters and present an explicit evaluation of the first few terms.

Counting the number of master integrals for sunrise diagrams via the Mellin-Barnes representation

“…In our case, the 5 Another example of such a cancellation was presented in ref. [37]. 6 We adopt the following definition of holonomic function [29]: a function is called holonomic if it satisfies a system of linear differential equations with polynomial coefficients whose solutions form a finite-dimensional vector space.…”

The elliptic dilogarithm for the sunset graph

“…An asymptotic expansion of the sunset integral has been given in [15]. Various forms for the integral have been considered either in geometrical terms [16], displaying some relations to one-loop amplitude [17], or a representation in terms of hypergeometric function as given in [18,19,20,21] or as an integral of Bessel functions as in [22,23]. Or a differential equation approach (in close relation to the method used in section 5 of the present work) was considered in [24,25,26,27,28].…”

A Feynman integral and its recurrences and associators

“…On the other hand, the expansion of the last function is expressible in terms of MPLs of q-th roots of unity times powers of logarithms ln(1 − z) [7]. Again, according to [7] any of the three functions (4.2) can be expressed in terms of the first one (4.1). Therefore, it is sufficient to study the ǫ-expansion of (4.1).…”

“…After introducing the new variable [1,7] with the q-th roots of unity (4.3). Then the system (4.7) can be brought into Fuchsian class…”

“…For a subclass of the latter the coefficients of their expansions generically represent multiple polylogarithms (MPLs), which give rise to periods of mixed Tate motives in algebraic geometry. Then, expansions around rational numbers p/q naturally yield q-th roots of unity exp(2πip/q) in the arguments of the MPLs [6,7].…”