Luise Adams

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

Simplifying Differential Equations for Multiscale Feynman Integrals beyond Multiple Polylogarithms

D=4-2\varepsilon

The ε-form of the differential equations for Feynman integrals in the elliptic case

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 sunrise integral and elliptic polylogarithms

In this Letter we exploit factorization properties of Picard-Fuchs operators to decouple differential equations for multiscale Feynman integrals. The algorithm reduces the differential equations to blocks of the size of the order of the irreducible factors of the Picard-Fuchs operator. As a side product, our method can be used to easily convert the differential equations for Feynman integrals which evaluate to multiple polylogarithms to an ϵ form.

A walk on sunset boulevard

Abstract Feynman integrals are easily solved if their system of differential equations is in e-form. In this letter we show by the explicit example of the kite integral family that an e-form can even be achieved, if the Feynman integrals do not evaluate to multiple polylogarithms. The e-form is obtained by a (non-algebraic) change of basis for the master integrals.

From elliptic curves to Feynman integrals

We summarize recent computations with a class of elliptic generalizations of polylogarithms, arising from the massive sunrise integral. For the case of arbitrary masses we obtain results in two and four space-time dimensions. The iterated integral structure of our functions allows us to furthermore compute the equal mass case to arbitrary order.