Related papers: Explicit solutions of the multi-loop integral recu…

We present a finite-order system of recurrence relations for a permanent of circulant matrices containing a band of k any-value diagonals on top of a uniform matrix (for k = 1, 2, and 3) as well as the method for deriving such recurrence…

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…

We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which…

We present a method to construct a suitable contour deformation in loop momentum space for multi-loop integrals. This contour deformation can be used to perform the integration for multi-loop integrals numerically. The integration can be…

The central idea of this article is to present a systematic approach to construct some recurrence relations for the solutions of the second-order linear difference equation of hypergeometric-type defined on the quadratic-type lattices. We…

A recurrence equation is a discrete integrable equation whose solutions are all periodic and the period is fixed. We show that infinitely many recurrence equations can be derived from the information about invariant varieties of periodic…

This paper provides details of the massless three-loop three-point integrals calculation at the symmetric point. Our work aimed to extend known two-loop results for such integrals to the three-loop level. Obtained results can find their…

I present a few new and recent ideas of the multiloop calculations.

We calculate 3-loop master integrals for heavy quark correlators and the 3-loop QCD corrections to the $\rho$-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be…

Recurrence relations derived via the Chetyrkin--Tkachov method of integration by parts are applied to reduce scalar three-loop bubble (vacuum) diagrams with a mass to a limited number of master integrals. The reduction is implemented as a…

Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…

New iterative methods for solving linear equations are presented that are easy to use, generalize good existing methods, and appear to be faster. The new algorithms mix two kinds of linear recurrence formulas. Older methods have either high…

A fully numerical method to calculate loop integrals, a numerical contour-integration method, is proposed. Loop integrals can be interpreted as a contour integral in a complex plane for an integrand with multi-poles in the plane. Stable and…

We derive the recurrence relations for relativistic Coulomb integrals directly from the integral representations with the help of computer algebra methods. In order to manage the computational complexity of this problem, we employ holonomic…

A new approach is presented to evaluate multi-loop integrals, which appear in the calculation of cross-sections in high-energy physics. It relies on a fully numerical method and is applicable to a wide class of integrals with various mass…

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 show that the problem of solving recurrence relations for L-loop (R+1)-point Feynman integrals within the method of integration by parts is equivalent to the corresponding problem for (L+R)-loop vacuum or (L+R-1)-loop propagator-type…

The paper is devoted to the methods of solving simultaneous recurrences. Specifically, we discuss transformation of matrix recurrences to regular recurrences and propose a way of solving special matrix recurrences of order three by their…

We review recent progress that we have achieved in evaluating the class of fully massive vacuum integrals at five loops. After discussing topics that arise in classification, evaluation and algorithmic codification of this specific set of…

This article is the second of a series of three presenting an alternative method to compute the one-loop scalar integrals. It extends the results of the first article to general complex masses. Let us remind the main features enjoyed by…