Christian Bogner

FEYNMAN GRAPH POLYNOMIALS

The integrand of any multiloop integral is characterized after Feynman parametrization by two polynomials. In this review we summarize the properties of these polynomials. Topics covered in this paper include among others: spanning trees and spanning forests, the all-minors matrix-tree theorem, recursion relations due to contraction and deletion of edges, Dodgsons identity and matroids.

The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms

We present the two-loop sunrise integral with arbitrary non-zero masses in two space-time dimensions in terms of elliptic dilogarithms. We find that the structure of the result is as simple and elegant as in the equal mass case, only the arguments of the elliptic dilogarithms are modified. These arguments have a nice geometric interpretation.

The two-loop sunrise graph with arbitrary masses

We discuss the analytical solution of the two-loop sunrise graph with arbitrary non-zero masses in two space-time dimensions. The analytical result is obtained by solving a second-order differential equation. The solution involves elliptic integrals and in particular the solutions of the corresponding homogeneous differential equation are given by periods of an elliptic curve.

The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case

We present the result for the finite part of the two-loop sunrise integral with unequal masses in four space-time dimensions in terms of the O(e0)-part and the O(e1)-part of the sunrise integral around two space-time dimensions. The latter two integrals are given in terms of elliptic generalisations of Clausen and Glaisher functions. Interesting aspects of the result for the O(e1)-part of the sunrise integral around two space-time dimensions are the occurrence of depth two elliptic objects and the weights of the individual terms.

The iterated structure of the all-order result for the two-loop sunrise integral

We present a method to compute the Laurent expansion of the two-loop sunrise integral with equal non-zero masses to arbitrary order in the dimensional regularisation e. This is done by introducing a class of functions (generalisations of multiple polylogarithms to include the elliptic case) and by showing that all integrations can be carried out within this class of functions.

The kite integral to all orders in terms of elliptic polylogarithms

We show that the Laurent series of the two-loop kite integral in

Symbolic integration and multiple polylogarithms

D=4-2\varepsilon

Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral

space-time dimensions can be expressed in each order of the series expansion in terms of elliptic generalisations of (multiple) polylogarithms. Using differential equations we present an iterative method to compute any desired order. As an example, we give the first three orders explicitly.

The real radiation antenna functions for S\to Q\overline{Q}gg at NNLO QCD

We review a method for the algebraic treatment of a family of functions which contains the multiple polylogarithms, with applications to the symbolic calculation of Feynman integrals.

The sunrise integral and elliptic polylogarithms

Abstract We study the analytic continuation of Feynman integrals from the kite family, expressed in terms of elliptic generalisations of (multiple) polylogarithms. Expressed in this way, the Feynman integrals are functions of two periods of an elliptic curve. We show that all what is required is just the analytic continuation of these two periods. We present an explicit formula for the two periods for all values of t ∈ R . Furthermore, the nome q of the elliptic curve satisfies over the complete range in t the inequality | q | ≤ 1 , where | q | = 1 is attained only at the singular points t ∈ { m 2 , 9 m 2 , ∞ } . This ensures the convergence of the q -series expansion of the ELi-functions and provides a fast and efficient evaluation of these Feynman integrals.