Related papers: Explicit solutions of the multi-loop integral recu…
We introduce and solve an infinite class of loop integrals which generalises the well-known ladder series. The integrals are described in terms of single-valued polylogarithmic functions which satisfy certain differential equations. The…
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,…
To entirely determine the resulting functions of one-loop integrals it is necessary to find the correct analytic continuation to all relevant kinematical regions. We argue that this continuation procedure may be performed in a general and…
We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of…
In this paper we describe a new method of calculation of master integrals based on the solution of systems of difference equations in one variable. An explicit example is given, and the generalization to arbitrary diagrams is described. As…
A set of recurrence relations for on-shell two-loop self-energy diagrams with one mass is presented, which allows to reduce the diagrams with arbitrary indices (powers of scalar propagators) to a set of the master integrals. The SHELL2…
In this paper, we present formula solutions of a family of difference equations of higher order. We discuss the periodic nature of the solutions and we investigate the stability character of the equilibrium points. We utilize Lie symmetry…
New solution method for the systems of linear equations in commutative integral domains is proposed. Its complexity is the same that the complexity of the matrix multiplication.
Two-term recurrence relations are supplied for indefinite integrals of functions that involve factors of the types ${P_2}^n$, ${P_3}^n$, ${P_4}^n$, ${P_1}^m {Q_1}^n$, $E_1 {P_1}^n$, ${P_1}^m {Q_2}^n$, $E_1 {P_2}^n$, ${P_2}^m {Q_2}^n$,…
We derive recurrence relations for the calculation of multiloop sunset-type diagrams with large powers of massive propagators. The technique is formulated in configuration space and exploits the explicit form of the massive propagator…
A numerical approach to compute tensor integrals in one-loop calculations is presented. The algorithm is based on a recursion relation which allows to express high rank tensor integrals as a function of lower rank ones. At each level of…
We provide an update on a long-term project that aims at evaluating massive vacuum integrals at the five-loop frontier, with high precision and in various space-time dimensions. A number of applications are sketched, mainly concerning the…
Planar L-loop maximally helicity violating amplitudes in N = 4 supersymmetric Yang-Mills theory are believed to possess the remarkable property of satisfying iteration relations in L. We propose a simple new method for studying the…
In modern quantum field theory, one of the most important tasks is the calculation of loop integrals. Loop integrals appear when evaluating the Feynman diagrams with one or more loops by integrating over the internal momenta. Even though…
We study matrix three term relations for orthogonal polynomials in two variables constructed from orthogonal polynomials in one variable. Using the three term recurrence relation for the involved univariate orthogonal polynomials, the…
New approach to systems of polynomial recursions is developed based on the Carleman linearization procedure. The article is divided into two main sections: firstly, we focus on the case of uni-variable depth-one polynomial recurrences.…
We study the applicability of the Z-Sum approach to multi-loop calculations with massive particles in perturbative quantum field theory. We systematically analyze the case of one-loop scalar integrals, which represent the building blocks of…
We show some of the mathematics that is being developed for the computation of deep inelastic structure functions to three loops. These include harmonic sums, harmonic polylogarithms and a class of difference equations that can be solved…
In this work, we present an algorithm for the diagonalization of the Integration-by-Parts (IBP) equations. Diagonalized IBP equations are indispensable for reducing loop integrals with high numerator powers to master integrals and for…
We describe a constructive procedure to separate overlapping infrared divergences in multi-loop integrals. Working with a parametric representation in D=4-2*epsilon dimensions, adequate subtractions lead to a Laurent series in epsilon,…