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We show how to compute the two-loop sunset integrals at finite volume, for non-degenerate masses and non-zero momentum. We present results for all integrals that appear in the Chiral Perturbation Therory ($\chi$PT) calculation of the…
A method for calculating phase-space master integrals for the decay process $1 \to n$ massless partons in QCD using integration-by-parts and differential equations techniques is discussed. The method is based on the appropriate choice of…
We consider a two-loop sunrise integral with two different internal masses at pseudo-threshold kinematics and we solve it in terms of elliptic polylogarithms to all orders of the dimensional regulator.
I describe a method to calculate a class of three-loop selfenergy diagrams for arbitrary internal masses and external momentum. This method combines analytical results and numerical integration, and is suitable for implementation in a…
In a previous paper a new approach has been introduced for computing, recursively and numerically, one-loop tensor integrals. Here we describe a few modifications of the original method that allow a more efficient numerical implementation…
We report on the development of tools to calculate loop integrals and amplitudes beyond one loop. In particular, we review new features of the program SecDec which can be used for the numerical evaluation of parametric integrals like…
We compute the master integrals that arise in the calculation of the leading penguin amplitudes in non-leptonic B-decays at two-loop order. The application of differential equations in a canonical basis enables us to give analytic results…
We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.
We review the Laporta algorithm for the reduction of scalar integrals to the master integrals and the differential equations technique for their evaluation. We discuss the use of the basis of harmonic polylogarithms for the analytical…
The variational iteration method is used to solve nonlinear Volterra integral equations. Two approaches are presented distinguished by the method to compute the Lagrange multiplier.
Starting from the parametric representation of a Feynman diagram, we obtain it's well defined value in dimensional regularisation by changing the integrals over parameters into contour integrals. That way we eventually arrive at a…
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…
At variance with fully inclusive quantities, which have been computed already at the two- or three-loop level, most exclusive observables are still known only at one-loop, as further progress was hampered so far by the greater computational…
In this paper we present a special formula for transforming integrals to series. The resulting series involves binomial transforms with the Taylor coefficients of the integrand. Five applications are provided for evaluating challenging…
Integral transformations are used to estimate high order derivatives of various special functions. Applications are given to numerical integration, where estimates of high order derivatives of the integrand are needed to achieve bounds on…
This contribution proposes a new formulation to efficiently compute directional derivatives of order one to fourth. The formulation is based on automatic differentiation implemented with dual numbers. Directional derivatives are particular…
An effective method to obtain exact analytical solutions of equations describing the coherent dynamics of multilevel systems is presented. The method is based on the usage of orthogonal polynomials, integral transforms and their discrete…
Reference [1] introduces a method for computing numerically four-dimensional multi-loop integrals without performing an explicit analytic contour deformation around threshold singularities. In this paper, we extend such a technique to…
The new version 2.1 of the program SecDec is described, which can be used for the factorisation of poles and subsequent numerical evaluation of multi-loop integrals, in particular massive two-loop integrals. The program is not restricted to…
In this paper, we develop a class of robust numerical methods for solving dynamical systems with multiple time scales. We first represent the solution of a multiscale dynamical system as a transformation of a slowly varying solution. Then,…