0000000000311454
AUTHOR
Christian Bogner
showing 14 related works from this author
Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral
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 $\mathr…
The kite integral to all orders in terms of elliptic polylogarithms
2016
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.
Resolution of singularities for multi-loop integrals
2007
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.
The sunrise integral and elliptic polylogarithms
2016
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.
A walk on sunset boulevard
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.
Analytic Continuation of the Kite Family
2019
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.
Differential equations for Feynman integrals beyond multiple polylogarithms
2017
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.
Blowing up Feynman integrals
2008
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.
The unequal mass sunrise integral expressed through iterated integrals on M‾1,3
2020
Abstract We solve the two-loop sunrise integral with unequal masses systematically to all orders in the dimensional regularisation parameter e. In order to do so, we transform the system of differential equations for the master integrals to an e-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 M 1 , 3 of a genus one Riemann surface with three marked points. This gives us the solution as iterated integrals on M ‾ 1 , 3 . On the hypersurface τ = const our result reduces to elliptic polylogarithms. In the equal mass case our result reduces to iterated integrals of modular forms.
Feynman graph polynomials
2010
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.
The Elliptic Sunrise
2020
In this talk, we discuss our recent computation of the two-loop sunrise integral with arbitrary non-zero particle masses in the vicinity of the equal mass point. 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.
The Elliptic Sunrise
2015
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.
The unequal mass sunrise integral expressed through iterated integrals on
2020
We solve the two-loop sunrise integral with unequal masses systematically to all orders in the dimensional regularisation parameter ε. In order to do so, we transform the system of differential equations for the master integrals to an ε-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 M1,3 of a genus one Riemann surface with three marked points. This gives us the solution as iterated integrals on M‾1,3. On the hypersurface τ=const our result reduces to elliptic polylogarithms. In the equal mass case our result reduces to iterated integrals of modular forms.
Resolution of singularities for multi-loop integrals
2019
Abstract 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. Title of program: sector_decomposition Catalogue Id: AEAG_v1_0 Nature of problem Computation of divergent multi-loop integrals. Versions of this program held in the CPC repository in Mendeley Data AEAG_v1_0; sect…