Hypergeometric Functions and Feynman Diagrams

Abstract

The relationship between Feynman diagrams and hypergeometric functions is discussed. Special attention is devoted to existing techniques for the construction of the -expansion. As an example, we present a detailed discussion of the construction of the -expansion of the Appell function 3 around rational values of parameters via an iterative solution of differential equations. 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 hypergeometric functions are briefly discussed.

Differential equations and Feynman integrals

“…I decided to reproduce these results using the IBP relations, which proved to be very 2 Sometimes it is convenient to stop the considered procedure on one-loop massive FI, the calculation of which can be performed, for example, by the method of Feynman parameters (see, for example, [9]. 3 Investigations of hypergeometric functions related to the calculation of FI are recently presented [10] as contribution to the Proceedings. 4 The results for massless diagrams are sometimes useful to obtain in x-space (see [22,23]).…”

Analytic integration methods in quantum field theory: an Introduction

“…In the case of three-loop ladder graphs also Appell-functions [81,82] contribute. A survey on the status of this method has been given in [8]. In relating the different special functions of this kind contiguous relations play an essential role, which has been discussed in [9] in detail.…”

“…The asymptotic representation of these quantities is thus given to arbitrary precision and one may use the recurrence relations of these quantities to analytically continue (8) from integer values of N to N ∈ C. Here it is important to observe the crossing relations for the respective process [214,215] which either implies the analytic continuation from the even or from the odd integers. These steps also apply to the other types of sums which were described in Refs.…”

“…The topics of the workshop included both techniques to reduce the number of Feynman diagrams by physical relations, such as the integration by parts relations [5][6][7] as well as the mathematical methods to compute these integrals analytically. The latter include the method of generalized hypergeometric functions [8] and the general theory of contiguous relations [9], the methods of integer relations [10], guessing methods of one-dimensional quantities, hyperlogarithms [11], the solution of master-integrals using difference and differential equations [12][13][14][15], Risch algorithms on nested integrals and rationalization algorithms [16], holonomic integration [17], the multivalued Almkvist-Zeilberger algorithm [18], expansion by regions [19], elliptic integrals and related topics [20,21], cutting techniques [22], and special multi-leg applications [23][24][25]. In different precision calculations these technologies are applied.…”

Olsson.wl : a Mathematica package for the computation of linear transformations of multivariable hypergeometric functions

“…As an example, multivariable hypergeometric functions arise in the context of perturbative calculations in quantum field theory. It seems that Regge was the first to show that Feynman integrals are linked to multivariable hypergeometric functions [2] (for a recent review of the relation between hypergeometric functions and Feynman integrals, we refer the reader to [3]). Nowadays, methods like the Mellin-Barnes (MB) representation [4,5], the method of brackets [6,7,8], the negative-dimension approach [9,10], etc.…”