92,938 research outputs found
Integration by parts identities in integer numbers of dimensions. A criterion for decoupling systems of differential equations
Integration by parts identities (IBPs) can be used to express large numbers
of apparently different d-dimensional Feynman Integrals in terms of a small
subset of so-called master integrals (MIs). Using the IBPs one can moreover
show that the MIs fulfil linear systems of coupled differential equations in
the external invariants. With the increase in number of loops and external
legs, one is left in general with an increasing number of MIs and consequently
also with an increasing number of coupled differential equations, which can
turn out to be very difficult to solve. In this paper we show how studying the
IBPs in fixed integer numbers of dimension d=n with one can
extract the information useful to determine a new basis of MIs, whose
differential equations decouple as and can therefore be more easily
solved as Laurent expansion in (d-n).Comment: 31 pages, minor typos corrected, references added, accepted for
publication in Nuclear Physics
Kira - A Feynman Integral Reduction Program
In this article, we present a new implementation of the Laporta algorithm to
reduce scalar multi-loop integrals---appearing in quantum field theoretic
calculations---to a set of master integrals. We extend existing approaches by
using an additional algorithm based on modular arithmetic to remove linearly
dependent equations from the system of equations arising from
integration-by-parts and Lorentz identities. Furthermore, the algebraic
manipulations required in the back substitution are optimized. We describe in
detail the implementation as well as the usage of the program. In addition, we
show benchmarks for concrete examples and compare the performance to Reduze 2
and FIRE 5.
In our benchmarks we find that Kira is highly competitive with these existing
tools.Comment: 37 pages, 3 figure
Reconstructing Rational Functions with
We present the open-source library for the
reconstruction of multivariate rational functions over finite fields. We
discuss the involved algorithms and their implementation. As an application, we
use in the context of integration-by-parts reductions and
compare runtime and memory consumption to a fully algebraic approach with the
program .Comment: 46 pages, 3 figures, 6 tables; v2: matches published versio
Calculation of Feynman integrals by difference equations
In this paper we describe a method of calculation of master integrals based
on the solution of systems of difference equations in one variable. Various
explicit examples are given, as well as the generalization to arbitrary
diagrams.Comment: LaTex, 10 pages, uses appolb.cls. Presented at the XXVII
International Conference of Theoretical Physics "Matter to the Deepest",
Ustron, Poland, 15-21 September 2003. To appear in Acta Physica Polonica.
v2:added reference
Conceptual modelling: Towards detecting modelling errors in engineering applications
Rapid advancements of modern technologies put high demands on mathematical modelling of engineering systems. Typically, systems are no longer âsimpleâ objects, but rather coupled systems involving multiphysics phenomena, the modelling of which involves coupling of models that describe different phenomena. After constructing a mathematical model, it is essential to analyse the correctness of the coupled models and to detect modelling errors compromising the final modelling result. Broadly, there are two classes of modelling errors: (a) errors related to abstract modelling, eg, conceptual errors concerning the coherence of a model as a whole and (b) errors related to concrete modelling or instance modelling, eg, questions of approximation quality and implementation. Instance modelling errors, on the one hand, are relatively well understood. Abstract modelling errors, on the other, are not appropriately addressed by modern modelling methodologies. The aim of this paper is to initiate a discussion on abstract approaches and their usability for mathematical modelling of engineering systems with the goal of making it possible to catch conceptual modelling errors early and automatically by computer assistant tools. To that end, we argue that it is necessary to identify and employ suitable mathematical abstractions to capture an accurate conceptual description of the process of modelling engineering systems
Computing periods of rational integrals
A period of a rational integral is the result of integrating, with respect to
one or several variables, a rational function over a closed path. This work
focuses particularly on periods depending on a parameter: in this case the
period under consideration satisfies a linear differential equation, the
Picard-Fuchs equation. I give a reduction algorithm that extends the
Griffiths-Dwork reduction and apply it to the computation of Picard-Fuchs
equations. The resulting algorithm is elementary and has been successfully
applied to problems that were previously out of reach.Comment: To appear in Math. comp. Supplementary material at
http://pierre.lairez.fr/supp/periods
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