相关论文: Feynman integrals and difference equations

We describe a new method of calculation of generic multi-loop master integrals based on the numerical solution of systems of difference equations in one variable. We show algorithms for the construction of the systems using…

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.

The computation of Feynman integrals is often the bottleneck of multi-loop calculations. We propose and implement a new method to efficiently evaluate such integrals in the physical region through the numerical integration of a suitable set…

We discuss a progress in calculation of Feynman integrals which has been done with help of the differential equation method and demonstrate the results for a class of two-point two-loop diagrams.

In this survey article we present difference field algorithms for symbolic summation. Special emphasize is put on new aspects in how the summation problems are rephrased in terms of difference fields, how the problems are solved there, and…

A large class of Feynman integrals, like e.g., two-point parameter integrals with at most one mass and containing local operator insertions, can be transformed to multi-sums over hypergeometric expressions. In this survey article we present…

We discuss a progress in calculation of Feynman integrals which has been done with help of the Differential Equation Method and demonstrate the results for a class of two-point two-loop diagrams.

The standard procedure for computing scalar multi-loop Feynman integrals consists in reducing them to a basis of so-called master integrals, derive differential equations in the external invariants satisfied by the latter and, finally, try…

Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations. These lectures give a review of these developments, while not assuming any prior knowledge of the…

Scattering amplitudes at loop level can be expressed in terms of Feynman integrals. The latter satisfy partial differential equations in the kinematical variables. We argue that a good choice of basis for (multi-)loop integrals can lead to…

Feynman integrals obey linear relations governed by intersection numbers, which act as scalar products between vector spaces. We present a general algorithm for constructing multivariate intersection numbers relevant to Feynman integrals,…

We present a Mathematica package AmpRed for the semi-automatic calculations of multi-loop Feynman amplitudes with high efficiency and precision. AmpRed implements the methods of integration by parts and differential equations in the…

In this paper, we describe a numerical approach to evaluate Feynman loop integrals. In this approach the key technique is a combination of a numerical integration method and a numerical extrapolation method. Since the computation is carried…

Using integration by parts relations, Feynman integrals can be represented in terms of coupled systems of differential equations. In the following we suppose that the unknown Feynman integrals can be given in power series representations,…

By carefully analyzing the relations between operator methods and the discretized and continuum path integral formulations of quantum-mechanical systems, we have found the correct Feynman rules for one-dimensional path integrals in curved…

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…

We describe a method to numerically compute multi-loop integrals, depending on one dimensionless parameter $x$ and the dimension $d$, in the whole kinematic range of $x$. The method is based on differential equations, which, however, do not…

We device a new method to calculate a large number of Mellin moments of single scale quantities using the systems of differential and/or difference equations obtained by integration-by-parts identities between the corresponding Feynman…

We show that tree-level and one-loop Mellin space correlators in anti-de Sitter space obey certain difference equations, which are the direct analog to the differential equations for Feynman loop integrals in the flat space.…

The method of differential equations has been proven to be a powerful tool for the computation of multi-loop Feynman integrals appearing in quantum field theory. It has been observed that in many instances a canonical basis can be chosen,…