Feynman diagrams and their algebraic lattices

Abstract

We present the lattice structure of Feynman diagram renormalization in physical QFTs from the viewpoint of Dyson-Schwinger-Equations and the core Hopf algebra of Feynman diagrams. The lattice structure encapsules the nestedness of diagrams. This structure can be used to give explicit expressions for the counterterms in zero-dimensional QFTs using the lattice-Moebius function. Different applications for the tadpole-free quotient, in which all appearing elements correspond to semimodular lattices, are discussed. * DK thanks the Alexander von Humboldt Foundation and the BMBF for support by an Alexander von Humboldt Professorship. It is a pleasure for both authors to thank David Broadhurst, Spencer Bloch, Dominique Manchon and Karen Yeats for helpful discussions. We also thank Fréderic Fauvet for organizing the workshop Resurgence, Physics and Numbers, Centro de Giorgi, Pisa, May 2015, and hospitality there.

A forest formula to subtract infrared singularities in amplitudes for wide-angle scattering

“…JHEP05(2020)012 9) as are shown in figures 36(c) and (d). The pinch surfaces of t σ 3 t σ 1 A are identical to those of t σ 3 A, because the only difference between them is to replace the partonic line in (b) by an eikonal line in the same direction in (c).…”

“…In principle, we should be able to also use equivalent forests [13] instead of forests to subtract IR divergences identically. These "freedoms" suggest a common and general mathematical structure (like the Hopf algebra [8,9]) in these different approaches.…”

“…Later, the BPHZ theorem was generalized to include massless fields by Lowenstein and Zimmermann [4,5], and also to include Euclidean infrared (IR) singularities by Chetyrkin, Tkachov and Smirnov [6,7]. Beyond this, a Hopf algebraic structure has been discovered by Kreimer [8,9], and its mathematical structures shed light on quantum field theory.…”

“…(2.4) and (2.5) are good approximations, we introduce the neighborhood of a pinch surface σ in terms of the coordinates in (2.1). This is defined as a region containing σ, where the normal coordinates s satisfy [17] 9) where p 2 0 Q 2 , and 0 < δ j < 1/2 is fixed for each intrinsic coordinate. The reason for this range of δ j is that if δ j < 1/2, we may always neglect l 2 and k 2 A terms on the r.h.s.…”

Feynman Diagrams beyond Physics: From Biology to Economy

“…The abstract nature of the mathematics behind Feynman diagrams suggest the wider use of these tools, which actually transcends the original physical domain. Indeed, we remark that the mathematical significance of Feynman diagrams is well studied in terms of algebraic lattice structures in the framework of Hopf algebras [72,73] and tensor models [74].…”