Related papers: Algebraic Algorithms in Perturbative Calculations
We study the core Hopf algebra underlying the renormalization Hopf algebra.
We give a Hopf-algebraic formulation of the $R^*$-operation, which is a canonical way to render UV and IR divergent Euclidean Feynman diagrams finite. Our analysis uncovers a close connection to Brown's Hopf algebra of motic graphs. Using…
We contruct here the Hopf algebra structure underlying the process of renormalization of non-commutative quantum field theory.
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 develop a generating-function formulation for the symbolic reduction of multi-loop Feynman integrals. In this framework, integration-by-parts identities are rewritten as differential equations for sector-wise generating functions, so the…
We describe the application of differential reduction algorithms for Feynman Diagram calculation. We illustrate the procedure in the context of generalized hypergeometric functions, and give an example for a type of q-loop bubble diagram.
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…
The hypergeometric function method naturally provides the analytic expressions of scalar integrals from concerned Feynman diagrams in some connected regions of independent kinematic variables, also presents the systems of homogeneous linear…
We extend Gegenbauer Polynomials technique to evaluate a class of complicated Feynman diagrams. New results in the form of $_3F_2$-hypergeometrical series of unit argument, are presented. As a by-product, we present a new transformation…
A method is presented in which matrix elements for some processes are calculated recursively. This recursive calculational technique is based on the method of basis spinors.
The present paper provides a method for finding partial differential equations satisfied by the Feynman integrals for diagrams of various types, using the Griffiths theorem on the reduction of poles of rational differential forms. As an…
We construct a Hopf algebra structure on the space of specified Feynman graphs of a quantum field theory. We introduce a convolution product and a semigroup of characters of this Hopf algebra with values in some suitable commutative algebra…
We give the solution of certain parabolic evolution problems (time-depending perturbations of the heat equation for the harmonic oscillator) as explicit integrals on the Wiener space.
In recent three--loop calculations of massive Feynman integrals within Quantum Chromodynamics (QCD) and, e.g., in recent combinatorial problems the so-called generalized harmonic sums (in short $S$-sums) arise. They are characterized by…
Operads and PROPs are presented, together with examples and applications to quantum physics suggesting the structure of Feynman categories/PROPs and the corresponding algebras.
We review the appearance of Hopf algebras in the renormalization of quantum field theories and in the study of diffeomorphisms of the frame bundle important for index computations in noncommutative geometry.
In this talk we present techniques for calculating one-loop amplitudes for multi-leg processes using Feynman diagrammatic methods in a semi-algebraic context. Our approach combines the advantages of the different methods allowing for a fast…
We propose that Kreimer's method of Feynman diagram renormalization via a Hopf algebra of rooted trees can be fruitfully employed in the analysis of block spin renormalization or coarse graining of inhomogeneous statistical systems.…
Single-scale Feynman diagrams yield integrals that are periods, namely projective integrals of rational functions of Schwinger parameters. Algebraic geometry may therefore inform us of the types of number to which these integrals evaluate.…
We consider Feynman integrals with algebraic leading singularities and total differentials in $\epsilon\,\mathrm{d}\ln$ form. We show for the first time that it is possible to evaluate integrals with singularities involving unrationalizable…