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We solve the two-loop sunrise integral with unequal masses systematically to all orders in the dimensional regularisation parameter $\varepsilon$. In order to do so, we transform the system of differential equations for the master integrals to an $\varepsilon$-form. The sunrise integral with unequal masses depends on three kinematical variables. We perform a change of variables to standard coordinates on the moduli space ${\mathcal M}_{1,3}$ of a genus one Riemann surface with three marked points. This gives us the solution as iterated integrals on $\overline{\mathcal M}_{1,3}$. On the hypersurface $\tau=\mbox{const}$ our result reduces to elliptic polylogarithms. In the equal mass case our result reduces to iterated integrals of modular forms., Comment: 34 pages, v2: version to be published

We give a brief introduction to a parametric approach for the derivation of shift relations between Feynman integrals and a result on the number of master integrals. The shift relations are obtained from parametric annihilators of the Lee-Pomeransky polynomial $\mathcal{G}$. By identification of Feynman integrals as multi-dimensional Mellin transforms, we show that this approach generates every shift relation. Feynman integrals of a given family form a vector space, whose finite dimension is naturally interpreted as the number of master integrals. This number is an Euler characteristic of the polynomial $\mathcal{G}$., Comment: Contribution to the proceedings of Loops and Legs in Quantum Field Theory (LL2018), 29 April - 04 May 2018, St. Goar (Germany)

We consider results for the master integrals of the kite family, given in terms of ELi-functions which are power series in the nome $q$ of an elliptic curve. The analytic continuation of these results beyond the Euclidean region is reduced to the analytic continuation of the two period integrals which define $q.$ We discuss the solution to the latter problem from the perspective of the Picard-Lefschetz formula., Comment: talk at the KMPB conference "Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory", 23-26 October 2017, at DESY Zeuthen

We study shift relations between Feynman integrals via the Mellin transform through parametric annihilation operators. These contain the momentum space IBP relations, which are well-known in the physics literature. Applying a result of Loeser and Sabbah, we conclude that the number of master integrals is computed by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate techniques to compute this Euler characteristic in various examples and compare it with numbers of master integrals obtained in previous works., Comment: v2: new section 3.1 added, several misprints corrected and additional remarks

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

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 $t \in {\mathbb R}$. Furthermore, the nome $q$ of the elliptic curve satisfies over the complete range in $t$ the inequality $|q|\le 1$, where $|q|=1$ is attained only at the singular points $t\in\{m^2,9m^2,\infty\}$. This ensures the convergence of the $q$-series expansion of the $\mathrm{ELi}$-functions and provides a fast and efficient evaluation of these Feynman integrals., Comment: 30 pages, 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

In this talk, we discuss recent progress in the application of generalizations of polylogarithms in the symbolic computation of multi-loop integrals. We briefly review the Maple program MPL which supports a certain approach for the computation of Feynman integrals in terms of multiple polylogarithms. Furthermore we discuss elliptic generalizations of polylogarithms which have shown to be useful in the computation of the massive two-loop sunrise integral., Comment: Talk presented at ACAT 2016 at UTFSM, Valpara\'iso, Chile

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

We introduce the computer program MPL for computations with homotopy invariant iterated integrals on moduli spaces $\mathcal{M}_{0,n}$ of curves of genus 0 with $n$ ordered marked points. The program is an implementation of the algorithms presented in [13], based on Maple. It includes the symbol map and procedures for the analytic computation of period integrals on $\mathcal{M}_{0,n}.$ It supports the automated computation of a certain class of Feynman integrals., Comment: 35 pages

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

This paper describes algorithms for the exact symbolic computation of period integrals on moduli spaces $\mathcal{M}_{0,n}$ of curves of genus $0$ with $n$ ordered marked points, and applications to the computation of Feynman integrals., Comment: 33 pages; minor corrections, references and an example added; version to be published

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 analyze, in the antenna subtraction framework, the real radiation antenna functions for processes involving the production of a pair of heavy quarks and two gluons by an uncolored initial state at NNLO QCD. We provide explicit expressions for these functions and discuss their infrared singular behaviour. Our main results are the corresponding integrated antenna functions which are computed analytically. They are expressed in terms of harmonic polylogarithms.

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

We review an approach for the computation of Feynman integrals by use of multiple polylogarithms, with an emphasis on the related criterion of linear reducibility of the graph. We show that the set of graphs which satisfies the linear reducibility with respect to both Symanzik polynomials is closed under taking minors. As a step towards a classification of Feynman integrals, we discuss the concept of critical minors and exhibit an example at three loops with four on-shell legs., Comment: 17 pages, to appear in the proceedings of the workshop "Periods and Motives", Madrid, July 2012

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., Comment: 8 pages, proceedings of the 11th DESY workshop "Loops and Legs in Quantum Field Theory", April 2012, Wernigerode, Germany

As a first step towards the application of the antenna subtraction formalism to NNLO QCD reactions with massive quarks, we determine the real radiation antenna function and its integrated counterpart for reactions of the type $S \to Q{\bar Q} q {\bar q}$, where $S$ denotes an uncolored initial state and $Q$, $q$ a massive and massless quark, respectively. We compute the corresponding integrated antenna function in terms of harmonic polylogarithms. As an application and check of our results we calculate the contribution proportional to $\alpha_s^2 e^2_Q N_f$ to the inclusive heavy-quark pair production cross ection in $e^+e^-$ annihilation., Comment: Latex, 32 pages

In this talk I discuss properties of the two Symanzik polynomials which characterise the integrand of an arbitrary multi-loop integral in its Feynman parametric form. Based on the construction from spanning forests and Laplacian matrices, Dodgson's relation is applied to derive factorisation identities involving both polynomials. An application of Whitney's 2-isomorphism theorem on matroids is discussed., Comment: Talk given at the International Workshop 'Loops and Legs in Quantum Field Theory' (April 25-30, 2010, W\"orlitz, Germany)

The integrand of any multi-loop integral is characterised after Feynman parametrisation by two polynomials. In this review we summarise the properties of these polynomials. Topics covered in this article include among others: Spanning trees and spanning forests, the all-minors matrix-tree theorem, recursion relations due to contraction and deletion of edges, Dodgson's identity and matroids., Comment: 35 pages, references added

In this talk we discuss mathematical structures associated to Feynman graphs. Feynman graphs are the backbone of calculations in perturbative quantum field theory. The mathematical structures -- apart from being of interest in their own right -- allow to derive algorithms for the computation of these graphs. Topics covered are the relations of Feynman integrals to periods, shuffle algebras and multiple polylogarithms., Comment: 24 pages, talk given at the conference "Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation", Pisa, October 2009

In this talk we discuss sector decomposition. This is a method to disentangle overlapping singularities through a sequence of blow-ups. We report on an open-source implementation of this algorithm to compute numerically the Laurent expansion of divergent multi-loop integrals. We also show how this method can be used to prove a theorem which relates the coefficients of the Laurent series of dimensionally regulated multi-loop integrals to periods., Comment: 6 pages, talk given at the conference "Loops and Legs in Quantum Field Theory", Sondershausen, April 2008

We consider multi-loop integrals in dimensional regularisation and the corresponding Laurent series. We study the integral in the Euclidean region and where all ratios of invariants and masses have rational values. We prove that in this case all coefficients of the Laurent series are periods., Comment: 22 pages, appendix added, version to be published

We report on a program for the numerical evaluation of divergent multi-loop integrals. The program is based on iterated sector decomposition. We improve the original algorithm of Binoth and Heinrich such that the program is guaranteed to terminate. The program can be used to compute numerically the Laurent expansion of divergent multi-loop integrals regulated by dimensional regularisation. The symbolic and the numerical steps of the algorithm are combined into one program., Comment: 23 pages, minor changes, version to be published