Related papers: Algorithms for the symbolic integration of hyperlo…
An algorithm for the systematic analytical approximation of multi-scale Feynman integrals is presented. The algorithm produces algebraic expressions as functions of the kinematical parameters and mass scales appearing in the Feynman…
In this talk I review the connections between Feynman integrals and multiple polylogarithms. After an introductory section on loop integrals I discuss the Mellin-Barnes transformation and shuffle algebras. In a subsequent section multiple…
We calculate convergent 3-loop Feynman diagrams containing a single massive loop equipped with twist $\tau =2$ local operator insertions corresponding to spin $N$. They contribute to the massive operator matrix elements in QCD describing…
In these lectures we discuss some of the mathematical structures that appear when computing multi-loop Feynman integrals. We focus on a specific class of special functions, the so-called multiple polylogarithms, and discuss introduce their…
Many interesting and useful symbolic computation algorithms manipulate mathematical expressions in mathematically meaningful ways. Although these algorithms are commonplace in computer algebra systems, they can be surprisingly difficult to…
We study Feynman integrals in the representation with Schwinger parameters and derive recursive integral formulas for massless 3- and 4-point functions. Properties of analytic (including dimensional) regularization are summarized and we…
The computation of Feynman integrals often involves square roots. One way to obtain a solution in terms of multiple polylogarithms is to rationalize these square roots by a suitable variable change. We present a program that can be used to…
This paper describes generalized polylogarithms, multiple polylogarithms, and multiple zeta values, along with their implementation in Maple 2018. This set of related functions is of interest in high energy physics as well as in number…
Given a Feynman parameter integral, depending on a single discrete variable $N$ and a real parameter $\epsilon$, we discuss a new algorithmic framework to compute the first coefficients of its Laurent series expansion in $\epsilon$. In a…
Integer relation algorithms can convert numerical results for Feynman integrals to exact evaluations, when one has reason to suspect the existence of reductions to linear combinations of a basis, with rational or algebraic coefficients.…
Symbolic integration deals with the evaluation of integrals in closed form. We present an overview of Risch's algorithm including recent developments. The algorithms discussed are suited for both indefinite and definite integration. They…
Symbolic summation as an active research topic of symbolic computation provides efficient algorithmic tools for evaluating and simplifying different types of sums arising from mathematics, computer science, physics and other areas. Most of…
We introduce a machine-learning framework based on symbolic regression to extract the full symbol alphabet of multi-loop Feynman integrals. By targeting the analytic structure rather than reduction, the method is broadly applicable and…
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…
Symbol letters are crucial for analytically calculating Feynman integrals in terms of iterated integrals. We present a novel method to construct the symbol letters for a given integral family without prior knowledge of the canonical…
Feynman integral reduction by means of integration-by-parts identities is a major power gadget in a theorist toolbox indispensable for calculation of multiloop quantum effects relevant for particle phenomenology and formal theory alike. An…
This paper presents a generalised symbolic algorithm for solving systems of linear algebraic equations with multi-diagonal coefficient matrices. The algorithm is given in a pseudocode. A theorem which gives the condition for correctness of…
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…
The paper deals with the process of mathematical modeling representations of exponential and logarithmic functions hypercomplex number system of generalized quaternions via determining a linear differential equation with hypercomplex…
Symmetry plays a central role in accelerating symbolic computation involving polynomials. This chapter surveys recent developments and foundational methods that leverage the inherent symmetries of polynomial systems to reduce complexity,…