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In this talk we discuss Feynman integrals which are related to elliptic curves. We show with the help of an explicit example that in the set of master integrals more than one elliptic curve may occur. The technique of maximal cuts is a useful tool to identify the elliptic curves. By a suitable transformation of the master integrals the system of differential equations for our example can be brought into a form linear in $\varepsilon$, where the $\varepsilon^0$-term is strictly lower-triangular. This system is easily solved in terms of iterated integrals., Comment: 11 pages, talk given at Loops and Legs 2018

In this talk we discuss a class of Feynman integrals, which can be expressed to all orders in the dimensional regularisation parameter as iterated integrals of modular forms. We review the mathematical prerequisites related to elliptic curves and modular forms. Feynman integrals, which evaluate to iterated integrals of modular forms go beyond the class of multiple polylogarithms. Nevertheless, we may bring for all examples considered the associated system of differential equations by a non-algebraic transformation to an $\varepsilon$-form, which makes a solution in terms of iterated integrals immediate., Comment: 21 pages, talk given at the KMPB conference "Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory"

In this article we give the details on the analytic calculation of the master integrals for the planar double box integral relevant to top-pair production with a closed top loop. We show that these integrals can be computed systematically to all order in the dimensional regularisation parameter $\varepsilon$. This is done by transforming the system of differential equations into a form linear in $\varepsilon$, where the $\varepsilon^0$-part is a strictly lower triangular matrix. Explicit results in terms of iterated integrals are presented for the terms relevant to NNLO calculations., Comment: 78 pages, additional file "supplementary_material.mpl" attached, v2: version to be published

We compute systematically for the planar double box Feynman integral relevant to top pair production with a closed top loop the Laurent expansion in the dimensional regularisation parameter $\varepsilon$. This is done by transforming the system of differential equations for this integral and all its sub-topologies to a form linear in $\varepsilon$, where the $\varepsilon^0$-part is strictly lower triangular. This system is easily solved order by order in the dimensional regularisation parameter $\varepsilon$. This is an example of an elliptic multi-scale integral involving several elliptic sub-topologies. Our methods are applicable to similar problems., Comment: 6 pages, version to be published

Feynman integrals are easily solved if their system of differential equations is in $\varepsilon$-form. In this letter we show by the explicit example of the kite integral family that an $\varepsilon$-form can even be achieved, if the Feynman integrals do not evaluate to multiple polylogarithms. The $\varepsilon$-form is obtained by a (non-algebraic) change of basis for the master integrals., Comment: 15 pages, version to be published

Differential equations are a powerful tool to tackle Feynman integrals. In this talk we discuss recent progress, where the method of differential equations has been applied to Feynman integrals which are not expressible in terms of multiple polylogarithms., Comment: 9 pages, talk given at RADCOR 2017

In this paper we show that certain Feynman integrals can be expressed as linear combinations of iterated integrals of modular forms to all orders in the dimensional regularisation parameter $\varepsilon$ . We discuss explicitly the equal mass sunrise integral and the kite integral. For both cases we give the alphabet of letters occurring in the iterated integrals. For the sunrise integral we present a compact formula, expressing this integral to all orders in $\varepsilon$ as iterated integrals of modular forms., Comment: 53 pages, v2: section on relation to Gamma_1(6) added, v3: section on the choice of the periods added

In this paper we exploit factorisation properties of Picard-Fuchs operators to decouple differential equations for multi-scale 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 $\varepsilon$-form., Comment: 5 pages, v2: version to be published

We show that the Laurent series of the two-loop kite integral in $D=4-2\varepsilon$ 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., Comment: 19 pages, v2: version to be published

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., Comment: talk given at Loops and Legs 2016

A walk on sunset boulevard can teach us about transcendental functions associated to Feynman diagrams. On this guided tour we will see multiple polylogarithms, differential equations and elliptic curves. A highlight of the tour will be the generalisation of the polylogarithms to the elliptic setting and the all-order solution for the sunset integral in the equal mass case., Comment: 10 pages, talk given at Radcor 2015

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 $\varepsilon$. 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., Comment: 20 pages, v2: version to be published

In this talk, we discuss our recent computation of the two-loop sunrise integral with arbitrary non-zero particle masses. In two space-time dimensions, we arrive at a result in terms of elliptic dilogarithms. Near four space-time dimensions, we obtain a result which furthermore involves elliptic generalizations of Clausen and Glaisher functions., Comment: talk at the Workshop on Multiple Zeta Values, Modular Forms and Elliptic Motives II, December 2014, ICMAT Madrid

In this talk we discuss the solution for the sunrise integral around two and four space-time dimensions in terms of a generalised elliptic version of the multiple polylogarithms. In two space-time dimensions we obtain a sum of three elliptic dilogarithms. The arguments of the elliptic dilogarithms have a nice geometric interpretation. In four space-time dimensions the sunrise integral can be expressed with the $\epsilon^0$- and $\epsilon^1$-solution around two dimensions, mass derivatives thereof and simpler terms., Comment: 5 pages, no figures, proceeding to the conference 'Matter To The Deepest', Sept 14-18, 2015, Ustro\'n, Poland

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 ${\mathcal O}(\varepsilon^0)$-part and the ${\mathcal O}(\varepsilon^1)$-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 ${\mathcal O}(\varepsilon^1)$-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., Comment: 31 pages

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., Comment: 22 pages, version to be published

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., Comment: 24 pages