Search results for "Symbolic"
showing 10 items of 449 documents
Products of Bessel functions and associated polynomials
2013
The symbolic method is used to get explicit formulae for the products or powers of Bessel functions and for the relevant integrals.
Speeding up a few orders of magnitude the Jacobi method: high order Chebyshev-Jacobi over GPUs
2017
In this technical note we show how to reach a remarkable speed up when solving elliptic partial differential equations with finite differences thanks to the joint use of the Chebyshev-Jacobi method with high order discretizations and its parallel implementation over GPUs.
Factorization of denominators in integration-by-parts reductions
2020
We present a Mathematica package which finds a basis of master integrals for the Feynman integral reduction. In this basis the dependence on the dimensional regularization in the denominators factorizes in kinematic independent polynomials.
Simplifying differential equations for multi-scale Feynman integrals beyond multiple polylogarithms
2017
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.
Simple differential equations for Feynman integrals associated to elliptic curves
2019
The $\varepsilon$-form of a system of differential equations for Feynman integrals has led to tremendeous progress in our abilities to compute Feynman integrals, as long as they fall into the class of multiple polylogarithms. It is therefore of current interest, if these methods extend beyond the case of multiple polylogarithms. In this talk I discuss Feynman integrals, which are associated to elliptic curves and their differential equations. I show for non-trivial examples how the system of differential equations can be brought into an $\varepsilon$-form. Single-scale and multi-scale cases are discussed.
Integral Reduction with Kira 2.0 and Finite Field Methods
2021
We present the new version 2.0 of the Feynman integral reduction program Kira and describe the new features. The primary new feature is the reconstruction of the final coefficients in integration-by-parts reductions by means of finite field methods with the help of FireFly. This procedure can be parallelized on computer clusters with MPI. Furthermore, the support for user-provided systems of equations has been significantly improved. This mode provides the flexibility to integrate Kira into projects that employ specialized reduction formulas, direct reduction of amplitudes, or to problems involving linear system of equations not limited to relations among standard Feynman integrals. We show…
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.
On the computation of intersection numbers for twisted cocycles
2020
Intersection numbers of twisted cocycles arise in mathematics in the field of algebraic geometry. Quite recently, they appeared in physics: Intersection numbers of twisted cocycles define a scalar product on the vector space of Feynman integrals. With this application, the practical and efficient computation of intersection numbers of twisted cocycles becomes a topic of interest. An existing algorithm for the computation of intersection numbers of twisted cocycles requires in intermediate steps the introduction of algebraic extensions (for example square roots), although the final result may be expressed without algebraic extensions. In this article I present an improvement of this algorith…
Computation of form factors in massless QCD with finite master integrals
2016
We present the bare one-, two-, and three-loop form factors in massless Quantum Chromodynamics as linear combinations of finite master integrals. Using symbolic integration, we compute their $\epsilon$ expansions and thereby reproduce all known results with an independent method. Remarkably, in our finite basis, only integrals with a less-than-maximal number of propagators contribute to the cusp anomalous dimensions. We report on indications of this phenomenon at four loops, including the result for a finite, irreducible, twelve-propagator form factor integral. Together with this article, we provide our automated software setup for the computation of finite master integrals.
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.